CBSE Class 12 Math R S Aggarwal Solution Chapter 24 Cross Or Vector Product Of Vectors

Class 12^th Maths R S Aggarwal Solution

Exercise 24

Question 1.

Find and , when

and

Answer:

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

and

Question 2.

Find and , when

and

Answer:

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 3.

Find and , when

and

Answer:

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 4.

Find and , when

and

Answer:

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 5.

Find and , when

and

Answer:

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 6.

Find λ if .

Answer:

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 7.

If and , find .

Verify that (i) and are perpendicular to each other

and (ii) and are perpendicular to each other.

Answer:

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

If and are perpendicular to each other then,

⇒

i.e.,

And in the similar way, we have,

Hence proved.

Question 8.

Find the value of:

i. ii. iii.

Answer:

The value of is, …

⇒

ii.

The value of is, … …

⇒

iii.

The value of is, … …

⇒

Question 9.

Find the unit vectors perpendicular to both and when

and

Answer:

Let be the vector which is perpendicular to then we have,

…where k is a scalor

Thus, we have r is a unit vector,

So,

We have,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 10.

Find the unit vectors perpendicular to both and when

and

Answer:

Let be the vector which is perpendicular to then we have,

…where k is a scalar

Thus, we have r is a unit vector,

So,

We have,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 11.

Find the unit vectors perpendicular to both and when

and

Answer:

Let be the vector which is perpendicular to then we have,

…where k is a scalar

Thus, we have r is a unit vector,

So,

We have,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 12.

Find the unit vectors perpendicular to both and when

and

Answer:

Let be the vector which is perpendicular to then we have,

…where k is a scalar

Thus, we have r is a unit vector,

So,

We have,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 13.

Find the unit vectors perpendicular to the plane of the vectors

and

Answer:

Let be the vector which is perpendicular to then we have,

…where k is a scalar

Thus, we have r is a unit vector,

So,

We have,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 14.

Find a vector of magnitude 6 which is perpendicular to both the vectors

and .

Answer:

Let be the vector which is perpendicular to then we have,

…where k is a scalar

Thus, we have r is vector of magnitude 6,

So,

We have,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Here, as r is of magnitude 6 thus,

k = 6,

Thus,

Question 15.

Find a vector of magnitude 5 units, perpendicular to each of the vectors

and, where and

Answer:

Let be the vector which is perpendicular to then we have,

…where k is a scalar

Thus, we have r is vector of magnitude 5,

So,

We have,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Here, as r is of magnitude 5 thus,

k = 5,

Thus,

Question 16.

Find an angle between two vectors and with magnitudes 1 and 2 respectively and .

Answer:

We are given that and .

And,

So we have,

|| =

⇒

Question 17.

If , and , find a vector which is perpendicular to both and and for which .

Answer:

Given that

Let be the vector which is perpendicular to then we have,

…where k is a scalar

We have,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Given that

⇒

Question 18.

If , and , find a vector which is perpendicular to both and and for which .

Answer:

Given that

Let be the vector which is perpendicular to then we have,

…where k is a scalar

We have,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Given that

⇒

Question 19.

Prove that , where θ is the angle between and .

Answer:

We know that |

And |

So,

Hence, proved.

Question 20.

Write the value of p for which and are parallel vectors.

Answer:

As the vectors are parallel vectors so,

Thus,

We have,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ Thus, p = .

Question 21.

Verify that , when

, and

Answer:

To verify

We need to prove L.H.S = R.H.S

L.H.S we have,

Given,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

RHS is

⇒

Thus, LHS = RHS.

Question 22.

Verify that , when

, and .

Answer:

To verify

We need to prove L.H.S = R.H.S

L.H.S we have,

Given, ,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

RHS is

⇒

Thus, LHS = RHS.

Question 23.

Find the area of the parallelogram whose adjacent sides are represented by the vectors:

and

Answer:

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 24.

Find the area of the parallelogram whose adjacent sides are represented by the vectors:

and

Answer:

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 25.

Find the area of the parallelogram whose adjacent sides are represented by the vectors:

and

Answer:

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 26.

Find the area of the parallelogram whose adjacent sides are represented by the vectors:

and

Answer:

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 27.

Find the area of the parallelogram whose diagonal are represented by the vectors

and

Answer:

The diagonals are

Thus,

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 28.

Find the area of the parallelogram whose diagonal are represented by the vectors

and

Answer:

The diagonals are

Thus,

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.

Area =

Here,

have
,

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 29.

Find the area of the parallelogram whose diagonal are represented by the vectors

and .

Answer:

The diagonals are

Thus,

The area of the parallelogram = , where a and b are vectors of it’s adjacent sides.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 30.

Find the area of the triangle whose two adjacent sides are determined by the vectors

and

Answer:

The area of the triangle = , where a and b are it’s adjacent sides vectors.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 31.

Find the area of the triangle whose two adjacent sides are determined by the vectors

and .

Answer:

The area of the triangle = , where a and b are it’s adjacent sides vectors.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 32.

Using vectors, find the area of ΔABC whose vertices are

A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5)

Answer:

Through the vertices we get the adjacent vectors as,

The area of the triangle = , where a and b are it’s adjacent sides vectors.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 33.

Using vectors, find the area of ΔABC whose vertices are

A(1, 2, 3), B(2, −1, 4) and C(4, 5, Δ1) ((considering Δ1 as 1 ))

Answer:

Through the vertices we get the adjacent vectors as,

The area of the triangle = , where a and b are it’s adjacent sides vectors.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 34.

Using vectors, find the area of ΔABC whose vertices are

A(3, −1, 2), B(1, −1, −3) and C(4, −3, 1)

Answer:

Through the vertices we get the adjacent vectors as,

The area of the triangle = , where a and b are it’s adjacent sides vectors.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 35.

Using vectors, find the area of ΔABC whose vertices are

A(1, −1, 2), B(2, 1, −1) and C(3, −1, 2).

Answer:

Through the vertices we get the adjacent vectors as,

The area of the triangle = , where a and b are it’s adjacent sides vectors.

Area =

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

⇒ sq units

Question 36.

Using vector method, show that the given points A, B, C are collinear:

A(3, −5, 1), B(−1, 0, 8) and C(7, −10, −6)

Answer:

Through the vertices we get the adjacent vectors as,

To prove that A, B, C are collinear we need to prove that

So,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 37.

Using vector method, show that the given points A, B, C are collinear:

A(6, −7, −1), B(2, −3, 1) and C(4, −5, 0).

Answer:

Through the vertices we get the adjacent vectors as,

To prove that A, B, C are collinear we need to prove that

So,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Thus, A, B and C are collinear.

Question 38.

Show that the point A, B, C with position vectors , and respectively are collinear.

Answer:

Through the vertices we get the adjacent vectors as,

To prove that A, B, C are collinear we need to prove that

So,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Thus, A, B and C are collinear.

Question 39.

Show that the points having position vectors are collinear, whatever be .

Answer:

Through the vertices we get the adjacent vectors as,

To prove that A, B, C are collinear we need to prove that

So,

Here,

have

Thus, substituting the values of ,

in equation (i) we get

⇒

Thus, A, B and C are collinear.

Question 40.

Show that the points having position vector , and are collinear, whatever be .

Answer:

We have,

Through the vertices we get the adjacent vectors as,

To prove that A, B, C are collinear we need to prove that

So,

Here,

have

Thus, substituting the values of ,

in equation (i) we get

⇒

Thus, A, B and C are collinear.

Question 41.

Find a unit vector perpendicular to the plane ABC, where the points A, B, C, are , and respectively.

Answer:

A unit vector perpendicular to the plane ABC will be,

Through the vertices we get the adjacent vectors as,

Here,

have

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 42.

If and then find .

Answer:

and

Then,

We have,

Here,

have
and

⇒

Thus, substituting the values of ,

in equation (i) we get

⇒

Question 43.

If , and , find .

Answer:

We have,

So,

⇒

Question 44.

If , and , find the angle between and .

Answer:

We have,

⇒

Chapter 24 Cross Or Vector Product Of Vectors

Class 12th Maths R S Aggarwal Solution

Exercise 24

Class 12^th Maths R S Aggarwal Solution