##### Class 12^{th} Mathematics Part Ii CBSE Solution

**Exercise 10.1**- Represent graphically a displacement of 40 km, 30° east of north.…
- Classify the following measures as scalars and vectors. (i) 10 kg (ii) 2 meters…
- Classify the following as scalar and vector quantities. (i) time period (ii)…
- In Fig 10.6 (a square), identify the following vectors. (i) Coinitial (ii)…
- Answer the following as true or false. (i) vector a and - vector a are…

**Exercise 10.2**- Compute the magnitude of the following vectors: vector a = i + j+k vector b = 2…
- Write two different vectors having same magnitude.
- Write two different vectors having same direction.
- Find the values of x and y so that the vectors 2 i+3 j x i+y j are equal.…
- Find the scalar and vector components of the vector with initial point (2, 1)…
- Find the sum of the vectors vector a = i-2 j + k , vector b = - 2 i+4 j+5 k ,…
- Find the unit vector in the direction of the vector vector a = i + j+2 k…
- Find the unit vector in the direction of vector vector pq where P and Q are the…
- For given vectors, vector a = 2 i - j+2 k , vector b = - i + j - k find the…
- Find a vector in the direction of vector 5 i - j+2 k which has magnitude 8…
- Show that the vectors 2 i-3 j+4 k , - 4 i+6 j-8 k are collinear.
- Find the direction cosines of the vector i+2 j+3 k
- Find the direction cosines of the vector joining the points A(1, 2, -3) and…
- Show that the vector i + j + k is equally inclined to the axes OX, OY and OZ.…
- Find the position vector of a point R which divides the line joining two…
- Find the position vector of the mid point of the vector joining the points…
- Show that the points A, B and C with position vectors, vector a = 3 i-4 j-4 k…
- In triangle ABC (Fig 10.18), which of the following is not true: (a) (b) (c)…
- vector a vector b are two collinear vectors, then which of the following are…

**Exercise 10.3**- Find the angle between two vectors vector a vector b with magnitudes √3 and 2…
- Find the angle between the vectors i-2 j+3 k 3 i-2 j + k
- Find the projection of the vector i - j on the vector i + j
- Find the projection of the vector i + 3 mathfrakj + 7 k on the vector 7 i - j+8…
- Show that each of the given three vectors is a unit vector: Also, show that…
- Find | vector a| | vector b| if (vector a + vector b) , (vector a - vector b) =…
- Evaluate the product (3 vector a-5 vector b) (2 vector a+7 vector b)…
- Find the magnitude of two vectors vector a vector b , having the same magnitude…
- Find | vector x| if for a unit vector vector a , (vector x - vector a) (vector…
- If vector a = 2 i-2 j+3 k , vector b = - i+2 j + k vector c = 3 i + j are such…
- Show that | vector a| vector b+| vector b| vector a is perpendicular to |…
- If vector a vector a = 0 and vector a vector b = 0 then what can be concluded…
- If vector a , vector b , vector c are unit vectors such that vector a + vector…
- If either vector vector a = vector 0 vector b = vector 0 then vector a vector…
- If the vertices A, B, C of a triangle ABC are (1, 2, 3), (-1, 0, 0), (0, 1,…
- Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear.…
- Show that the vectors 2 i - j + k , i-3 j-5 k 3 i-4 j-4 k form the vertices of…
- If vector a is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then…

**Exercise 10.4**- Find | vector a x vector b| if vector a = i - j j+7 k vector b = 3 i-2 j+2 k…
- Find a unit vector perpendicular to each of the vector
- If a unit vector vector a makes angles pi /3 i , pi /4 j and an acute angle θ…
- Show that (vector a - vector b) x (vector a + vector b) = 2 (vector a x vector…
- Find λ and μ if (2 i+6 j+27 k) x (i + lambda j + μ k) = vector 0
- Given that vector a vector b = 0 vector a x vector b = vector 0 What can you…
- Let the vectors vector a , vector b , vector c be given as Then show that…
- If either vector a = vector 0 vector b = vector 0 then vector a x vector b =…
- Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5,…
- Find the area of the parallelogram whose adjacent sides are determined by the…
- Let the vectors vector a vector b be such that | vector a| = 3 | vector b| =…
- Area of a rectangle having vertices A, B, C and D with position vectors - i +…

**Miscellaneous Exercise**- Write down a unit vector in XY-plane, making an angle of 30° with the positive…
- Find the scalar components and magnitude of the vector joining the points P(x1,…
- A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of…
- If | vector a| = | vector b|+| vector c| , then is it true that | vector a| = |…
- Find the value of x for which x (i + j + k) is a unit vector.
- Find a vector of magnitude 5 units, and parallel to the resultant of the…
- If vector a = i + j + k , vector b = 2 i - j+3 k , and vector c = i-2 j + k ,…
- Show that the points A(1, - 2, - 8), B(5, 0, -2) and C(11, 3, 7) are collinear,…
- Find the position vector of a point R which divides the line joining two points…
- The two adjacent sides of a parallelogram are 2 i-4 j+5 k and i-2 j-3 k . Find…
- Show that the direction cosines of a vector equally inclined to the axes OX,…
- Let vector a = i + 4 j + 2 k , vector b = 3 i-2 j+7 k and vector c = 2 i - j+4…
- The scalar product of the vector i + j + k a unit vector along the sum of…
- If vector a , vector b , vector c are mutually perpendicular vectors of equal…
- Prove that (vector a + vector b) (vector a + vector b) = | vector a|^2 + |…
- If theta is the angle between two vectors vector a and vector b , then vector…
- Let vector a and vector b be two unit vectors and q is the angle between them.…
- The value of isA. 0 B. -1 C. 1 D. 3
- If θ is the angle between any two vectors vector a and vector b , then |…

**Exercise 10.1**

- Represent graphically a displacement of 40 km, 30° east of north.…
- Classify the following measures as scalars and vectors. (i) 10 kg (ii) 2 meters…
- Classify the following as scalar and vector quantities. (i) time period (ii)…
- In Fig 10.6 (a square), identify the following vectors. (i) Coinitial (ii)…
- Answer the following as true or false. (i) vector a and - vector a are…

**Exercise 10.2**

- Compute the magnitude of the following vectors: vector a = i + j+k vector b = 2…
- Write two different vectors having same magnitude.
- Write two different vectors having same direction.
- Find the values of x and y so that the vectors 2 i+3 j x i+y j are equal.…
- Find the scalar and vector components of the vector with initial point (2, 1)…
- Find the sum of the vectors vector a = i-2 j + k , vector b = - 2 i+4 j+5 k ,…
- Find the unit vector in the direction of the vector vector a = i + j+2 k…
- Find the unit vector in the direction of vector vector pq where P and Q are the…
- For given vectors, vector a = 2 i - j+2 k , vector b = - i + j - k find the…
- Find a vector in the direction of vector 5 i - j+2 k which has magnitude 8…
- Show that the vectors 2 i-3 j+4 k , - 4 i+6 j-8 k are collinear.
- Find the direction cosines of the vector i+2 j+3 k
- Find the direction cosines of the vector joining the points A(1, 2, -3) and…
- Show that the vector i + j + k is equally inclined to the axes OX, OY and OZ.…
- Find the position vector of a point R which divides the line joining two…
- Find the position vector of the mid point of the vector joining the points…
- Show that the points A, B and C with position vectors, vector a = 3 i-4 j-4 k…
- In triangle ABC (Fig 10.18), which of the following is not true: (a) (b) (c)…
- vector a vector b are two collinear vectors, then which of the following are…

**Exercise 10.3**

- Find the angle between two vectors vector a vector b with magnitudes √3 and 2…
- Find the angle between the vectors i-2 j+3 k 3 i-2 j + k
- Find the projection of the vector i - j on the vector i + j
- Find the projection of the vector i + 3 mathfrakj + 7 k on the vector 7 i - j+8…
- Show that each of the given three vectors is a unit vector: Also, show that…
- Find | vector a| | vector b| if (vector a + vector b) , (vector a - vector b) =…
- Evaluate the product (3 vector a-5 vector b) (2 vector a+7 vector b)…
- Find the magnitude of two vectors vector a vector b , having the same magnitude…
- Find | vector x| if for a unit vector vector a , (vector x - vector a) (vector…
- If vector a = 2 i-2 j+3 k , vector b = - i+2 j + k vector c = 3 i + j are such…
- Show that | vector a| vector b+| vector b| vector a is perpendicular to |…
- If vector a vector a = 0 and vector a vector b = 0 then what can be concluded…
- If vector a , vector b , vector c are unit vectors such that vector a + vector…
- If either vector vector a = vector 0 vector b = vector 0 then vector a vector…
- If the vertices A, B, C of a triangle ABC are (1, 2, 3), (-1, 0, 0), (0, 1,…
- Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear.…
- Show that the vectors 2 i - j + k , i-3 j-5 k 3 i-4 j-4 k form the vertices of…
- If vector a is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then…

**Exercise 10.4**

- Find | vector a x vector b| if vector a = i - j j+7 k vector b = 3 i-2 j+2 k…
- Find a unit vector perpendicular to each of the vector
- If a unit vector vector a makes angles pi /3 i , pi /4 j and an acute angle θ…
- Show that (vector a - vector b) x (vector a + vector b) = 2 (vector a x vector…
- Find λ and μ if (2 i+6 j+27 k) x (i + lambda j + μ k) = vector 0
- Given that vector a vector b = 0 vector a x vector b = vector 0 What can you…
- Let the vectors vector a , vector b , vector c be given as Then show that…
- If either vector a = vector 0 vector b = vector 0 then vector a x vector b =…
- Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5,…
- Find the area of the parallelogram whose adjacent sides are determined by the…
- Let the vectors vector a vector b be such that | vector a| = 3 | vector b| =…
- Area of a rectangle having vertices A, B, C and D with position vectors - i +…

**Miscellaneous Exercise**

- Write down a unit vector in XY-plane, making an angle of 30° with the positive…
- Find the scalar components and magnitude of the vector joining the points P(x1,…
- A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of…
- If | vector a| = | vector b|+| vector c| , then is it true that | vector a| = |…
- Find the value of x for which x (i + j + k) is a unit vector.
- Find a vector of magnitude 5 units, and parallel to the resultant of the…
- If vector a = i + j + k , vector b = 2 i - j+3 k , and vector c = i-2 j + k ,…
- Show that the points A(1, - 2, - 8), B(5, 0, -2) and C(11, 3, 7) are collinear,…
- Find the position vector of a point R which divides the line joining two points…
- The two adjacent sides of a parallelogram are 2 i-4 j+5 k and i-2 j-3 k . Find…
- Show that the direction cosines of a vector equally inclined to the axes OX,…
- Let vector a = i + 4 j + 2 k , vector b = 3 i-2 j+7 k and vector c = 2 i - j+4…
- The scalar product of the vector i + j + k a unit vector along the sum of…
- If vector a , vector b , vector c are mutually perpendicular vectors of equal…
- Prove that (vector a + vector b) (vector a + vector b) = | vector a|^2 + |…
- If theta is the angle between two vectors vector a and vector b , then vector…
- Let vector a and vector b be two unit vectors and q is the angle between them.…
- The value of isA. 0 B. -1 C. 1 D. 3
- If θ is the angle between any two vectors vector a and vector b , then |…

###### Exercise 10.1

**Question 1.**Represent graphically a displacement of 40 km, 30° east of north.

**Answer:**North (N), South (S), East (E), and West (W) are plotted on the paper as shown.

Let be the displacement vector such that = 40 km

Vector makes an angle of 30° east of North i.e. 30° with North in North-East direction.

**Question 2.**Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 meters north-west

(iii) 40° (iv) 40 watt

(v) 10^{-19} coulomb (vi) 20 m/s^{2}

**Answer:**Scalars are defined as quantities that have magnitude only.

Vectors are defined as a quantity that has magnitude as well as direction.

(i) **10 kg:** It is a measure of mass. It is a __scalar__ quantity as it has magnitude only and no direction.

(ii) **2 meters north-west:** It is a measure of distance in a particular direction. ∴ It is a __vector__ quantity as it has magnitude as well as direction.

(iii) **40°:** It is a measure of angle. It is a __scalar__ quantity as it has magnitude only and no direction.

(iv) **40 Watt:** It is a measure of power. It is a __scalar__ quantity as it has magnitude only and no direction.

(v) **10**^{-19} coulomb: It is a measure of electric charge. It is a __scalar__ quantity as it has magnitude only and no direction.

(vi) **20 m/sec**^{2}: It is a measure of acceleration. It is a __vector__ quantity as it is a measure of rate of change of velocity.

**Question 3.**Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

**Answer:**Scalars are defined as quantities that have magnitude only.

Vectors are defined as a quantity that has magnitude as well as direction.

(i) **time period:** It is a __scalar__ quantity as it has magnitude only and no direction.

(ii) **distance:** It is a __scalar__ quantity as it has magnitude only and no direction.

(iii) **force:** It is a __vector__ quantity as it has magnitude as well as direction.

(iv) **velocity:** It is a __vector__ quantity as it has magnitude as well as direction.

(v) **work done:** It is a __scalar__ quantity as it has magnitude as well as direction.

**Question 4.**In Fig 10.6 (a square), identify the following vectors.

(i) Coinitial

(ii) Equal

(iii) Collinear but not equal

**Answer:**

(i) **Coinitial vectors:** Two or more vectors having same initial point are called coinitial vectors.

So, in the above figure and are coinitial vectors as they have same initial point.

∴ Coinitial vectors: and

(ii) **Equal vectors:** Two or more vectors having same direction and same magnitude are called equal vectors.

So, in the above figure and are equal vectors as they have same magnitude and same direction.

∴ Equal vectors: and

(iii) **Collinear but not equal:**

∵ and are parallel vectors, so, they are collinear. But they have opposite direction, so, they are not equal.

Hence, and are collinear but not equal.

**Question 5.**Answer the following as true or false.

(i) and are collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

**Answer:**(i) and are collinear: True

Explanation: and are parallel vectors, so, they are collinear.

(ii) Two collinear vectors are always equal in magnitude: False

Explanation: we know, and are unit vectors, so they have same magnitude but and are vectors along x – axis and y – axis respectively.

(iii) Two vectors having same magnitude are collinear: False

Explanation: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.

We know, and are unit vectors, so they have same magnitude but and are vectors along x – axis and y – axis respectively. Since, they are not parallel to each other, so, they are not collinear.

(iv) Two collinear vectors having the same magnitude are equal: False

Explanation: Two vector are said to be equal, if they have the same magnitude and direction.

and are collinear vectors as they are parallel to each other and also they have same magnitude. But they do not have same direction, so, they are unequal vectors.

**Question 1.**

Represent graphically a displacement of 40 km, 30° east of north.

**Answer:**

North (N), South (S), East (E), and West (W) are plotted on the paper as shown.

Let be the displacement vector such that = 40 km

Vector makes an angle of 30° east of North i.e. 30° with North in North-East direction.

**Question 2.**

Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 meters north-west

(iii) 40° (iv) 40 watt

(v) 10^{-19} coulomb (vi) 20 m/s^{2}

**Answer:**

Scalars are defined as quantities that have magnitude only.

Vectors are defined as a quantity that has magnitude as well as direction.

(i) **10 kg:** It is a measure of mass. It is a __scalar__ quantity as it has magnitude only and no direction.

(ii) **2 meters north-west:** It is a measure of distance in a particular direction. ∴ It is a __vector__ quantity as it has magnitude as well as direction.

(iii) **40°:** It is a measure of angle. It is a __scalar__ quantity as it has magnitude only and no direction.

(iv) **40 Watt:** It is a measure of power. It is a __scalar__ quantity as it has magnitude only and no direction.

(v) **10 ^{-19} coulomb:** It is a measure of electric charge. It is a

__scalar__quantity as it has magnitude only and no direction.

(vi) **20 m/sec ^{2}:** It is a measure of acceleration. It is a

__vector__quantity as it is a measure of rate of change of velocity.

**Question 3.**

Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done

**Answer:**

Scalars are defined as quantities that have magnitude only.

Vectors are defined as a quantity that has magnitude as well as direction.

(i) **time period:** It is a __scalar__ quantity as it has magnitude only and no direction.

(ii) **distance:** It is a __scalar__ quantity as it has magnitude only and no direction.

(iii) **force:** It is a __vector__ quantity as it has magnitude as well as direction.

(iv) **velocity:** It is a __vector__ quantity as it has magnitude as well as direction.

(v) **work done:** It is a __scalar__ quantity as it has magnitude as well as direction.

**Question 4.**

In Fig 10.6 (a square), identify the following vectors.

(i) Coinitial

(ii) Equal

(iii) Collinear but not equal

**Answer:**

(i) **Coinitial vectors:** Two or more vectors having same initial point are called coinitial vectors.

So, in the above figure and are coinitial vectors as they have same initial point.

∴ Coinitial vectors: and

(ii) **Equal vectors:** Two or more vectors having same direction and same magnitude are called equal vectors.

So, in the above figure and are equal vectors as they have same magnitude and same direction.

∴ Equal vectors: and

(iii) **Collinear but not equal:**

∵ and are parallel vectors, so, they are collinear. But they have opposite direction, so, they are not equal.

Hence, and are collinear but not equal.

**Question 5.**

Answer the following as true or false.

(i) and are collinear.

(ii) Two collinear vectors are always equal in magnitude.

(iii) Two vectors having same magnitude are collinear.

(iv) Two collinear vectors having the same magnitude are equal.

**Answer:**

(i) and are collinear: True

Explanation: and are parallel vectors, so, they are collinear.

(ii) Two collinear vectors are always equal in magnitude: False

Explanation: we know, and are unit vectors, so they have same magnitude but and are vectors along x – axis and y – axis respectively.

(iii) Two vectors having same magnitude are collinear: False

Explanation: Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.

We know, and are unit vectors, so they have same magnitude but and are vectors along x – axis and y – axis respectively. Since, they are not parallel to each other, so, they are not collinear.

(iv) Two collinear vectors having the same magnitude are equal: False

Explanation: Two vector are said to be equal, if they have the same magnitude and direction.

and are collinear vectors as they are parallel to each other and also they have same magnitude. But they do not have same direction, so, they are unequal vectors.

###### Exercise 10.2

**Question 1.**

**Answer:**

**Question 2.**Write two different vectors having same magnitude.

**Answer:**Let

And

We can see clearly because the all the coefficients of and are not same. In and , the coefficients are different. Now, we check the magnitude of both,

And

So, magnitude of both vectors is same but they are different.

**Question 3.**Write two different vectors having same direction.

**Answer:**Let

And

So, here where m = 2 > 0

Therefore, both vectors have same direction.

Now, we check for the magnitude,

So, they have same direction but same magnitude.

**Question 4.**Find the values of x and y so that the vectors are equal.

**Answer:**For two vectors to be equal, the coefficients of both vectors should be equal.

Comparing the -coefficient, we get x=2

Comparing the -coefficient, we get y=3

**Question 5.**Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (– 5, 7).

**Answer:**

Let A be the initial point (2,1) and B be the final point (-5,7).

So, and

Now we want to find the

Therefore, the scalar components of are -7 and 6. The vector components of are .

**Question 6.**Find the sum of the vectors

**Answer:**

And

We want to find

To find the sum, we add coefficients of to each other.

So,

**Question 7.**Find the unit vector in the direction of the vector

**Answer:**We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as

So, we find the magnitude of first.

**Question 8.**Find the unit vector in the direction of vector where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

**Answer:**So, and

Now we want to find the

Now, we have to find the unit direction in the direction of .We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as

So, we find the magnitude of first.

**Question 9.**For given vectors, find the unit vector in the direction of the vector

**Answer:**

We want to find

To find the sum, we add coefficients of to each other.

So,

Now, we have to find the unit direction in the direction of.We know that unit vector means that the magnitude of the vector is 1(unit).

So, we find the magnitude of first.

**Question 10.**Find a vector in the direction of vector which has magnitude 8 units.

**Answer:**

Let

The vector in the direction of having unit magnitude is .

So, The vector in the direction of having magnitude 8 units =.

**Question 11.**Show that the vectors are collinear.

**Answer:**

We know that two vectors are collinear if they have the same direction or are parallel or anti-parallel. They can be expressed in the form where a and b are vectors and 'm' is a scalar quantity.

From the question,

Let and

Here,

So, where, m=-2.

∴ The given two vectors are collinear.

**Question 12.**Find the direction cosines of the vector

**Answer:**

The direction cosines of a vector are defined as the coefficients of in the unit vector in the direction of the vector.

So, first we find the unit vector in the direction of the vector.

Let

Therefore, The direction cosines of the given vector are .

**Question 13.**Find the direction cosines of the vector joining the points A(1, 2, –3) and B(–1, –2, 1), directed from A to B.

**Answer:**So, and

Now we want to find the

Now, we have to find the direction cosines which are the coefficients of the unit vector in the direction of .We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as

So, we find the magnitude of first.

Therefore, The direction cosines of the given vector are .

**Question 14.**Show that the vector is equally inclined to the axes OX, OY and OZ.

**Answer:**To find the inclination of the vector with OX, OY, OZ. We find the direction cosines of the vector.

We know that the direction cosines of a vector are defined as the coefficients of in the unit vector in the direction of the vector.

So, first we find the unit vector in the direction of the vector.

Let

Therefore, The direction cosines of the given vector are .

Let be the angle between and OX.

Therefore,

Similarly, Let be the angle between and OY.

Therefore,

And, be the angle between and OZ.

Therefore,

Therefore,

Hence proved that the vector is equally inclined with the axes OX, OY, and OZ.

**Question 15.**Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are respectively, in the ratio 2 : 1

(i) internally (ii) externally

**Answer:**Given that and

(i) The lies on the segment PQ(internal division).If m:n is the ratio in which divides PQ, then

Given m:n=2:1, m=2 and n=1

(ii) The does not lie on the segment PQ(external division).If m:n is the ratio in which divides PQ,then

Given m:n=2:1, m=2 and n=1

**Question 16.**Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, –2).

**Answer:**So, and

Let be the mid point of .

**Question 17.**Show that the points A, B and C with position vectors, respectively form the vertices of a right angled triangle.

**Answer:**O be the origin,

Let

And

Now we find the vectors

……….(1)

………….(2)

……….(3)

Adding equations (1), (2) and (3)

Therefore, ABC form a triangle.

Now, we want to prove that it is a right angled triangle.

Therefore, the triangle satisfies the Pythagoras theorem. Hence, proved the given vectors form a right angled triangle.

**Question 18.**In triangle ABC (Fig 10.18), which of the following is not true:

(a)

(b)

(c)

(d)

**Answer:**We know by triangle law of vectors,

…………..(1)

Therefore,

Hence (B) is true.

We know that

Putting in eq(1), we get

…….(2)

Hence (A) is also true.

Now,

Putting in eq(2)

Hence (D) is correct.

We are asked the option which is not true.

Therefore, the correct option is (C).

**Question 19.** are two collinear vectors, then which of the following are incorrect:

(a)

(b)

(c) the respective components of are not proportional

(d) both the vectors have same direction, but different magnitudes.

**Answer:**

We know that two vectors are collinear if they have the same direction or are parallel or anti-parallel. They can be expressed in the form where a and b are vectors and ' m ' is a scalar quantity.

Therefore, (a) is true.

In (b),

So, (b) is also true.

The vectors and are proportional,

Therefore, (c) is not true.

The vectors and can have different magnitude as well as different direction.

Therefore, (d) is not true.

We are asked the options which are not true.

∴ The correct answer is (c) and (d).

**Question 1.**

**Answer:**

**Question 2.**

Write two different vectors having same magnitude.

**Answer:**

Let

And

We can see clearly because the all the coefficients of and are not same. In and , the coefficients are different. Now, we check the magnitude of both,

And

So, magnitude of both vectors is same but they are different.

**Question 3.**

Write two different vectors having same direction.

**Answer:**

Let

And

So, here where m = 2 > 0

Therefore, both vectors have same direction.

Now, we check for the magnitude,

So, they have same direction but same magnitude.

**Question 4.**

Find the values of x and y so that the vectors are equal.

**Answer:**

For two vectors to be equal, the coefficients of both vectors should be equal.

Comparing the -coefficient, we get x=2

Comparing the -coefficient, we get y=3

**Question 5.**

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (– 5, 7).

**Answer:**

Let A be the initial point (2,1) and B be the final point (-5,7).

So, and

Now we want to find the

Therefore, the scalar components of are -7 and 6. The vector components of are .

**Question 6.**

Find the sum of the vectors

**Answer:**

And

We want to find

To find the sum, we add coefficients of to each other.

So,

**Question 7.**

Find the unit vector in the direction of the vector

**Answer:**

We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as

So, we find the magnitude of first.

**Question 8.**

Find the unit vector in the direction of vector where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

**Answer:**

So, and

Now we want to find the

Now, we have to find the unit direction in the direction of .We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as

So, we find the magnitude of first.

**Question 9.**

For given vectors, find the unit vector in the direction of the vector

**Answer:**

We want to find

To find the sum, we add coefficients of to each other.

So,

Now, we have to find the unit direction in the direction of.We know that unit vector means that the magnitude of the vector is 1(unit).

So, we find the magnitude of first.

**Question 10.**

Find a vector in the direction of vector which has magnitude 8 units.

**Answer:**

Let

The vector in the direction of having unit magnitude is .

So, The vector in the direction of having magnitude 8 units =.

**Question 11.**

Show that the vectors are collinear.

**Answer:**

We know that two vectors are collinear if they have the same direction or are parallel or anti-parallel. They can be expressed in the form where a and b are vectors and 'm' is a scalar quantity.

From the question,

Let and

Here,

So, where, m=-2.

∴ The given two vectors are collinear.

**Question 12.**

Find the direction cosines of the vector

**Answer:**

The direction cosines of a vector are defined as the coefficients of in the unit vector in the direction of the vector.

So, first we find the unit vector in the direction of the vector.

Let

Therefore, The direction cosines of the given vector are .

**Question 13.**

Find the direction cosines of the vector joining the points A(1, 2, –3) and B(–1, –2, 1), directed from A to B.

**Answer:**

So, and

Now we want to find the

Now, we have to find the direction cosines which are the coefficients of the unit vector in the direction of .We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as

So, we find the magnitude of first.

Therefore, The direction cosines of the given vector are .

**Question 14.**

Show that the vector is equally inclined to the axes OX, OY and OZ.

**Answer:**

To find the inclination of the vector with OX, OY, OZ. We find the direction cosines of the vector.

We know that the direction cosines of a vector are defined as the coefficients of in the unit vector in the direction of the vector.

So, first we find the unit vector in the direction of the vector.

Let

Therefore, The direction cosines of the given vector are .

Let be the angle between and OX.

Therefore,

Similarly, Let be the angle between and OY.

Therefore,

And, be the angle between and OZ.

Therefore,

Therefore,

Hence proved that the vector is equally inclined with the axes OX, OY, and OZ.

**Question 15.**

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are respectively, in the ratio 2 : 1

(i) internally (ii) externally

**Answer:**

Given that and

(i) The lies on the segment PQ(internal division).If m:n is the ratio in which divides PQ, then

Given m:n=2:1, m=2 and n=1

(ii) The does not lie on the segment PQ(external division).If m:n is the ratio in which divides PQ,then

Given m:n=2:1, m=2 and n=1

**Question 16.**

Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, –2).

**Answer:**

So, and

Let be the mid point of .

**Question 17.**

Show that the points A, B and C with position vectors, respectively form the vertices of a right angled triangle.

**Answer:**

O be the origin,

Let

And

Now we find the vectors

……….(1)

………….(2)

……….(3)

Adding equations (1), (2) and (3)

Therefore, ABC form a triangle.

Now, we want to prove that it is a right angled triangle.

Therefore, the triangle satisfies the Pythagoras theorem. Hence, proved the given vectors form a right angled triangle.

**Question 18.**

In triangle ABC (Fig 10.18), which of the following is not true:

(a)

(b)

(c)

(d)

**Answer:**

We know by triangle law of vectors,

…………..(1)

Therefore,

Hence (B) is true.

We know that

Putting in eq(1), we get

…….(2)

Hence (A) is also true.

Now,

Putting in eq(2)

Hence (D) is correct.

We are asked the option which is not true.

Therefore, the correct option is (C).

**Question 19.**

are two collinear vectors, then which of the following are incorrect:

(a)

(b)

(c) the respective components of are not proportional

(d) both the vectors have same direction, but different magnitudes.

**Answer:**

We know that two vectors are collinear if they have the same direction or are parallel or anti-parallel. They can be expressed in the form where a and b are vectors and ' m ' is a scalar quantity.

Therefore, (a) is true.

In (b),

So, (b) is also true.

The vectors and are proportional,

Therefore, (c) is not true.

The vectors and can have different magnitude as well as different direction.

Therefore, (d) is not true.

We are asked the options which are not true.

∴ The correct answer is (c) and (d).

###### Exercise 10.3

**Question 1.**Find the angle between two vectors with magnitudes √3 and 2 respectively having

**Answer:**Given,

We know,

Putting the value of

∴

So, the angle between the vectors is π/4

**Question 2.**

**Answer:**Given vectors are

Let

=

∴ We know,

⇒

⇒

⇒ cos θ = 10/14

⇒ θ = cos^{-1}(5/7)

**Question 3.**Find the projection of the vector on the vector

**Answer:**Let and

Projection of is given by

Now

∴ Projection of

= 0

∴ Projection of is 0.

**Question 4.**Find the projection of the vector on the vector

**Answer:**Let

Projection of is given by

Now

∴ Projection of

∴ Projection of .

**Question 5.**Show that each of the given three vectors is a unit vector:

Also, show that they are mutually perpendicular to each other.

**Answer:**Let

Now we need to find out the magnitude of

Since,

∴ the three given vectors are unit vectors.

To show that each of three vectors are mutually perpendicular to each other

We have to show

Since,

∴ are mutually perpendicular to each other.

**Question 6.**Find if

**Answer:**Given,

Now,

∴

⇒

∴

**Question 7.**Evaluate the product

**Answer:**To find

**Question 8.**Find the magnitude of two vectors , having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2.

**Answer:**Let θ be the angle between

Given,

and θ = 60°

Scalar product of

i.e

We know

∴ [magnitude of vector is positive]

So,

Hence,

**Question 9.**Find if for a unit vector

**Answer:**Given, vector is a unit vector

∴

∴

**Question 10.**If are such that is perpendicular to then find the value of λ.

**Answer:**Given,

It is given is perpendicular to , then

∴ λ = 8

**Question 11.**Show that is perpendicular to for any two nonzero vectors .

**Answer:**To show is perpendicular to for any two non zero vectors , we need to show dot product of is zero.

= 0

Since, dot product of is zero.

∴ is perpendicular to

**Question 12.**If and then what can be concluded about the vector

**Answer:**Given,

⇒

⇒

∴ is a zero vector.

Hence, can be any vector.

**Question 13.**If are unit vectors such that find the value of

**Answer:**Given are unit vectors and

Now

Adding (i), (ii) and (iii) we get

∴

**Question 14.**If either vector then But the converse need not be true. Justify your answer with an example.

**Answer:**If either = 0

Now let

So,

∴

∴

Since

Hence, the converse of given statement need not be true.

**Question 15.**If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC, [∠ABC is the angle between the vectors ].

**Answer:**Given points are A (1, 2, 3), B (-1, 0, 0) and C (0, 1, 2)

∠ABC is the angle between vectors

We know

⇒ ∠ABC = cos^{-1}(10/√102)

**Question 16.**Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.

**Answer:**Given points are A (1, 2, 7), B (2, 6, 3) and C (3, 10, -1)

So,

Since,

∴ The given points A, B and C are collinear.

**Question 17.**Show that the vectors form the vertices of a right angled triangle.

**Answer:**Let vectors be position vectors of points A, B and C respectively.

Then,

Now vectors represent sides of ΔABC

Now

And

∴

From Pythagoras theorem we know

If then ΔABC is a right-angled triangle

∴ Δ ABC is a right-angled triangle.

**Question 18.**If is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then is unit vector if

A. λ = 1

B. λ = –1

C. a = |λ|

D. a = 1/|λ|

**Answer:**Given λ is a unit vector.

And

Since

∴

λ is a unit vector if a = 1/|λ|

**Question 1.**

Find the angle between two vectors with magnitudes √3 and 2 respectively having

**Answer:**

Given,

We know,

Putting the value of

∴

So, the angle between the vectors is π/4

**Question 2.**

**Answer:**

Given vectors are

Let

=

∴ We know,

⇒

⇒

⇒ cos θ = 10/14

⇒ θ = cos^{-1}(5/7)

**Question 3.**

Find the projection of the vector on the vector

**Answer:**

Let and

Projection of is given by

Now

∴ Projection of

= 0

∴ Projection of is 0.

**Question 4.**

Find the projection of the vector on the vector

**Answer:**

Let

Projection of is given by

Now

∴ Projection of

∴ Projection of .

**Question 5.**

Show that each of the given three vectors is a unit vector:

Also, show that they are mutually perpendicular to each other.

**Answer:**

Let

Now we need to find out the magnitude of

Since,

∴ the three given vectors are unit vectors.

To show that each of three vectors are mutually perpendicular to each other

We have to show

Since,

∴ are mutually perpendicular to each other.

**Question 6.**

Find if

**Answer:**

Given,

Now,

∴

⇒

∴

**Question 7.**

Evaluate the product

**Answer:**

To find

**Question 8.**

Find the magnitude of two vectors , having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2.

**Answer:**

Let θ be the angle between

Given,

and θ = 60°

Scalar product of

i.e

We know

∴ [magnitude of vector is positive]

So,

Hence,

**Question 9.**

Find if for a unit vector

**Answer:**

Given, vector is a unit vector

∴

∴

**Question 10.**

If are such that is perpendicular to then find the value of λ.

**Answer:**

Given,

It is given is perpendicular to , then

∴ λ = 8

**Question 11.**

Show that is perpendicular to for any two nonzero vectors .

**Answer:**

To show is perpendicular to for any two non zero vectors , we need to show dot product of is zero.

= 0

Since, dot product of is zero.

∴ is perpendicular to

**Question 12.**

If and then what can be concluded about the vector

**Answer:**

Given,

⇒

⇒

∴ is a zero vector.

Hence, can be any vector.

**Question 13.**

If are unit vectors such that find the value of

**Answer:**

Given are unit vectors and

Now

Adding (i), (ii) and (iii) we get

∴

**Question 14.**

If either vector then But the converse need not be true. Justify your answer with an example.

**Answer:**

If either = 0

Now let

So,

∴

∴

Since

Hence, the converse of given statement need not be true.

**Question 15.**

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC, [∠ABC is the angle between the vectors ].

**Answer:**

Given points are A (1, 2, 3), B (-1, 0, 0) and C (0, 1, 2)

∠ABC is the angle between vectors

We know

⇒ ∠ABC = cos^{-1}(10/√102)

**Question 16.**

Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, –1) are collinear.

**Answer:**

Given points are A (1, 2, 7), B (2, 6, 3) and C (3, 10, -1)

So,

Since,

∴ The given points A, B and C are collinear.

**Question 17.**

Show that the vectors form the vertices of a right angled triangle.

**Answer:**

Let vectors be position vectors of points A, B and C respectively.

Then,

Now vectors represent sides of ΔABC

Now

And

∴

From Pythagoras theorem we know

If then ΔABC is a right-angled triangle

∴ Δ ABC is a right-angled triangle.

**Question 18.**

If is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then is unit vector if

A. λ = 1

B. λ = –1

C. a = |λ|

D. a = 1/|λ|

**Answer:**

Given λ is a unit vector.

And

Since

∴

λ is a unit vector if a = 1/|λ|

###### Exercise 10.4

**Question 1.**

**Answer:**Given that

and

Expanding along first row,

**Question 2.**Find a unit vector perpendicular to each of the vector

**Answer:**

Given that

and

Let

Let

Now, we want to find a vector which is perpendicular to both and . It is given by .

Let

Expanding along first row,

Therefore,

Now we want to find the unit vector. We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as

So, we find the magnitude of first.

**Question 3.**If a unit vector makes angles and an acute angle θ with then find θ and hence, the components of .

**Answer:**

Let

Given that it is a unit vector,

So, ……(1)

Let be the angle with respectively.

Then,

(given)

(given)

Putting value of x,y, z in equation (1)

(Squaring both sides)

As should be acute,

Components of are the coefficients of , which are,

**Question 4.**Show that

**Answer:**We solve for the left-hand side,

L.H.S.=

We know and

Also,

Putting these in our equation, we get

which is equal to R.H.S.

Hence proved.

**Question 5.**Find λ and μ if

**Answer:**Let

and

Given that

Expanding along first row,

By comparing the y-coefficients,

By comparing the z-coefficients,

These values should also satisfy the equation we will get from comparing the x-coefficients,

Therefore, .

**Question 6.**Given that What can you conclude about the vectors ?

**Answer:**Given that

(where θ is the angle between the vectors)

Also given that,

(where θ is the angle between the vectors)

As there is no value of θ for which both sin θ and cos θ are zero.

Therefore, the condition for which and is:

.

**Question 7.**Let the vectors be given as Then show that

**Answer:**Given that

and

Solving for left hand side,

First, we calculate

(Property of determinant)

Which is equal to R.H.S.

Hence proved.

**Question 8.**If either then Is the converse true? Justify your answer with an example.

**Answer:** (where θ is the angle between the vectors)

If

Similarly, If

Now, If

(where θ is the angle between the vectors)

sin θ = 0, this implies θ = 0°

This implies that the converse is not always true. The vectors may not be zero but the angle between is 0°, i.e. the vectors are parallel.

**Question 9.**Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

**Answer:**So,

Now we want to find the

Now we want to find the

We know that

So, we find first.

Expanding along first row,

**Question 10.**Find the area of the parallelogram whose adjacent sides are determined by the vectors

**Answer:**

Given

and

We know that

So, we find ,

Expanding along first row,

**Question 11.**Let the vectors be such that then is a unit vector, if the angle between is

A. π/6

B. π/4

C. π/3

D. π/2

**Answer:**

Given that:

(Magnitude of unit vector is 1)

Putting the values in the equation (1), we get

(where θ is the angle between the vectors)

Therefore, The correct option is (B).

**Question 12.**Area of a rectangle having vertices A, B, C and D with position vectors respectively is

A. 1/2

B. 1

C. 2

D. 4

**Answer:**Let

Now we want to find the

Now we want to find the

We know that

So, we find ,

Expanding along first row,

**Question 1.**

**Answer:**

Given that

and

Expanding along first row,

**Question 2.**

Find a unit vector perpendicular to each of the vector

**Answer:**

Given that

and

Let

Let

Now, we want to find a vector which is perpendicular to both and . It is given by .

Let

Expanding along first row,

Therefore,

Now we want to find the unit vector. We know that unit vector means that the magnitude of the vector is 1(unit).

It is defined as

So, we find the magnitude of first.

**Question 3.**

If a unit vector makes angles and an acute angle θ with then find θ and hence, the components of .

**Answer:**

Let

Given that it is a unit vector,

So, ……(1)

Let be the angle with respectively.

Then,

(given)

(given)

Putting value of x,y, z in equation (1)

(Squaring both sides)

As should be acute,

Components of are the coefficients of , which are,

**Question 4.**

Show that

**Answer:**

We solve for the left-hand side,

L.H.S.=

We know and

Also,

Putting these in our equation, we get

which is equal to R.H.S.

Hence proved.

**Question 5.**

Find λ and μ if

**Answer:**

Let

and

Given that

Expanding along first row,

By comparing the y-coefficients,

By comparing the z-coefficients,

These values should also satisfy the equation we will get from comparing the x-coefficients,

Therefore, .

**Question 6.**

Given that What can you conclude about the vectors ?

**Answer:**

Given that

(where θ is the angle between the vectors)

Also given that,

(where θ is the angle between the vectors)

As there is no value of θ for which both sin θ and cos θ are zero.

Therefore, the condition for which and is:

.

**Question 7.**

Let the vectors be given as Then show that

**Answer:**

Given that

and

Solving for left hand side,

First, we calculate

(Property of determinant)

Which is equal to R.H.S.

Hence proved.

**Question 8.**

If either then Is the converse true? Justify your answer with an example.

**Answer:**

(where θ is the angle between the vectors)

If

Similarly, If

Now, If

(where θ is the angle between the vectors)

sin θ = 0, this implies θ = 0°

This implies that the converse is not always true. The vectors may not be zero but the angle between is 0°, i.e. the vectors are parallel.

**Question 9.**

Find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

**Answer:**

So,

Now we want to find the

Now we want to find the

We know that

So, we find first.

Expanding along first row,

**Question 10.**

Find the area of the parallelogram whose adjacent sides are determined by the vectors

**Answer:**

Given

and

We know that

So, we find ,

Expanding along first row,

**Question 11.**

Let the vectors be such that then is a unit vector, if the angle between is

A. π/6

B. π/4

C. π/3

D. π/2

**Answer:**

Given that:

(Magnitude of unit vector is 1)

Putting the values in the equation (1), we get

(where θ is the angle between the vectors)

Therefore, The correct option is (B).

**Question 12.**

Area of a rectangle having vertices A, B, C and D with position vectors respectively is

A. 1/2

B. 1

C. 2

D. 4

**Answer:**

Let

Now we want to find the

Now we want to find the

We know that

So, we find ,

Expanding along first row,

###### Miscellaneous Exercise

**Question 1.**Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

**Answer:**Given: A unit vector in XY- plane.

Let is a unit vector in the given XY- plane then the value of

Will be :

θ is the angle given, which is made by unit vector with positive direction of x-axis.

∴ for θ = 30°

Hence, the required unit vector is .

**Question 2.**Find the scalar components and magnitude of the vector joining the points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}).

**Answer:**Given: points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are given.

The vector obtained by joining the given points P and Q:

= position vector of Q – position vector of P

Hence the scalar component of the vector obtained by joining the points are

[(x_{2}-x_{1}), (y_{2}-y_{1}), (z_{2}-z_{1})]

And the magnitude of the vector obtained by joining the points is

**Question 3.**A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

**Answer:**let O be the initial position and B be the final position of the girl.

Position of girl will be as shown in figure:

Hence, and ∠ BAO = 60°

Now, by using triangle law of vector addition,

Hence girls displacement from initial to final position is :

**Question 4.**

**Answer:**Let in given triangle

Now, by using triangle law of vector addition,

As we can see that represent the sides of triangle.

Also we know that sum of two sides of a triangle must be greater than its third side.

is not true.

**Question 5.**Find the value of x for which is a unit vector.

**Answer:**Given: as a unit vector.

Now, if is a unit vector then

**Question 6.**Find a vector of magnitude 5 units, and parallel to the resultant of the vectors and

**Answer:**Given:

Let the resultant of

Then

Then

Now the vector of magnitude 5 units and parallel to is:

**Question 7.**If , and , find a unit vector parallel to the vector .

**Answer:**Given:

Let the resultant of

Then

Then

∴ unit vector along

**Question 8.**Show that the points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.

**Answer:**Given: A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7)

Then

**Question 1.**

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

**Answer:**

Given: A unit vector in XY- plane.

Let is a unit vector in the given XY- plane then the value of

Will be :

θ is the angle given, which is made by unit vector with positive direction of x-axis.

∴ for θ = 30°

Hence, the required unit vector is .

**Question 2.**

Find the scalar components and magnitude of the vector joining the points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}).

**Answer:**

Given: points P(x_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) are given.

The vector obtained by joining the given points P and Q:

= position vector of Q – position vector of P

Hence the scalar component of the vector obtained by joining the points are

[(x_{2}-x_{1}), (y_{2}-y_{1}), (z_{2}-z_{1})]

And the magnitude of the vector obtained by joining the points is

**Question 3.**

A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

**Answer:**

let O be the initial position and B be the final position of the girl.

Position of girl will be as shown in figure:

Hence, and ∠ BAO = 60°

Now, by using triangle law of vector addition,

Hence girls displacement from initial to final position is :

**Question 4.**

**Answer:**

Let in given triangle

Now, by using triangle law of vector addition,

As we can see that represent the sides of triangle.

Also we know that sum of two sides of a triangle must be greater than its third side.

is not true.

**Question 5.**

Find the value of x for which is a unit vector.

**Answer:**

Given: as a unit vector.

Now, if is a unit vector then

**Question 6.**

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors and

**Answer:**

Given:

Let the resultant of

Then

Then

Now the vector of magnitude 5 units and parallel to is:

**Question 7.**

If , and , find a unit vector parallel to the vector .

**Answer:**

Given:

Let the resultant of

Then

Then

∴ unit vector along

**Question 8.**

Show that the points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.

**Answer:**

Given: A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7)

Then