RS Aggarwal - Mathematics

Book: RS Aggarwal - Mathematics

Chapter: 25. Product of Three Vectors

Subject: Maths - Class 12th

Listen NCERT Audio Books to boost your productivity and retention power by 2X.

Chapter Exercises

Exercise 25A
Exercise 25B
Objective Questions

Exercise 25A

1

Prove that

i.


ii.


3

Find when

i. and ..


ii. and


iii. and


3

Find the volume of the parallelepiped whose conterminous edges are represented by the vectors

i.


ii.


iii.


iv.


4

Show that the vectors are coplanar, when

i. and


ii. and


iii. and


5

Find the value of λ for which the vectors are coplanar, when

i. and


ii. and


iii. and


6

If and find and interpret the result.
7

The volume of the parallelepiped whose edges are and is 546 cubic units. Find the value of λ.
8

Show that the vectors and are parallel to the same plane.

{HINT: Show that }
9

If the vectors and be coplanar, show that c2 = ab.
10

Show that the four points with position vectors and are coplanar.
11

Show that the four points with position vectors and are coplanar.
12

Find the value of λ for which the four points with position vectors and are coplanar.

Ans. λ = 3
13

Find the value of λ for which the four points with position vectors and are coplanar.
14

Using vector method, show that the points A(4, 5, 1), B(0, -1, -1), C(3, 9, 4) and
D(-4, 4, 4) are coplanar.
15

Find the value of λ for which the points A(3, 2, 1), B(4, λ, 5), C(4, 2, -2) and D(6, 5, -1) are coplanar.

Ans. λ = 5

Exercise 25B

1

If and are two equal vectors the x + y + z = ?
2

Write a unit vector in the direction of the sum of the vectors and

3

Write the value of λ so that the vectors and are perpendicular to each other.
4

Find the value of p for which the vectors and are parallel.
5

Find the value of λ when the projection of on is 4 units.
6

If and are perpendicular vectors such that and find the value of

7

If is a unit vector such that find

8

Find the sum of the vectors and

9

Find the sum of the vectors and

10

Write the projection of the vector along the vector

11

Write the projection of the vector on the vector

12

Find when and

13

Find a vector in the direction of which has magnitude 21 units.
14

If and are such that is perpendicular to then find the value of λ.
15

Write the vector of magnitude 15 units in the direction of vector

16

If and find a vector of magnitude 6 units which is parallel to the vector

17

Write the projection of the vector on the vector

18

Write the angle between two vectors and with magnitudes and 2 respectively having

19

If and then find

20

Find the angle between two vectors and with magnitudes 1 and 2 respectively, when

21

What conclusion can you draw about vectors and when and

22

Find the value of λ when the vectors and are parallel.
23

Write the value of
24

Find the volume of the parallelepiped whose edges are represented by the vectors and

25

If and then prove that and are coplanar.
26

If and are such that then find the values of λ and μ.
27

If θ is the angle between and and then what is the value of θ?
28

When does hold?
29

Find the direction cosines of a vector which is equally inclined to the x - axis, y - axis and z - axis.
30

If P(1, 5, 4) and Q(4, 1, - 2) be the position vectors of two points P and Q, find the direction ratios of

31

Find the direction cosines of the vector .
32

If and are unit vectors such that is a unit vector, what is the angle between and ?
1

If and are two equal vectors the x + y + z = ?
2

Write a unit vector in the direction of the sum of the vectors and

3

Write the value of λ so that the vectors and are perpendicular to each other.
4

Find the value of p for which the vectors and are parallel.
5

Find the value of λ when the projection of on is 4 units.
6

If and are perpendicular vectors such that and find the value of

7

If is a unit vector such that find

8

Find the sum of the vectors and

9

Find the sum of the vectors and

10

Write the projection of the vector along the vector

11

Write the projection of the vector on the vector

12

Find when and

13

Find a vector in the direction of which has magnitude 21 units.
14

If and are such that is perpendicular to then find the value of λ.
15

Write the vector of magnitude 15 units in the direction of the vector

16

If and find a vector of magnitude 6 units which is parallel to the vector

17

Write the projection of the vector on the vector

18

Write the angle between two vectors and with magnitudes and 2 respectively having

19

If and then find

20

Find the angle between two vectors and with magnitudes 1 and 2 respectively, when

21

What conclusion can you draw about vectors and when and

22

Find the value of λ when the vectors and are parallel.
23

Write the value of
24

Find the volume of the parallelepiped whose edges are represented by the vectors and

25

If and then prove that and are coplanar.
26

If and are such that then find the values of λ and μ.
27

If θ is the angle between and and then what is the value of θ?
28

When does hold?
29

Find the direction cosines of a vector which is equally inclined to the x - axis, y - axis and z - axis.
30

If P(1, 5, 4) and Q(4, 1, - 2) be the position vectors of two points P and Q, find the direction ratios of

31

Find the direction cosines of the vector .
32

If and are unit vectors such that is a unit vector, what is the angle between and ?

Objective Questions

1

Mark (√) against the correct answer in each of the following:

A unit vector in the direction of the vector is
1

Mark (√) against the correct answer in each of the following:

A unit vector in the direction of the vector is
2

Mark (√) against the correct answer in each of the following:

The direction cosines of the vector are
2

Mark (√) against the correct answer in each of the following:

The direction cosines of the vector are
3

Mark (√) against the correct answer in each of the following:

If A(1, 2, -3) and B(-1, -2, 1) are the end points of a vector then the direction cosines of are
3

Mark (√) against the correct answer in each of the following:

If A(1, 2, -3) and B(-1, -2, 1) are the end points of a vector then the direction cosines of are
4

Mark (√) against the correct answer in each of the following:

If a vector makes angle α, β and γ with the x-axis, y-axis and z-axis respectively then the value of (sin2α + sin2β + sin2γ) is
4

Mark (√) against the correct answer in each of the following:

If a vector makes angle α, β and γ with the x-axis, y-axis and z-axis respectively then the value of (sin2α + sin2β + sin2γ) is
5

Mark (√) against the correct answer in each of the following:

The vector is a
5

Mark (√) against the correct answer in each of the following:

The vector is a
6

Mark (√) against the correct answer in each of the following:

What is the angle which the vector makes with the z-axis?
6

Mark (√) against the correct answer in each of the following:

What is the angle which the vector makes with the z-axis?
7

Mark (√) against the correct answer in each of the following:

f and are vectors such that and then the angle between and is
7

Mark (√) against the correct answer in each of the following:

f and are vectors such that and then the angle between and is
8

Mark (√) against the correct answer in each of the following:

If and are two vectors such that and then the angle between and is
8

Mark (√) against the correct answer in each of the following:

If and are two vectors such that and then the angle between and is
9

Mark (√) against the correct answer in each of the following:

The angle between the vectors and is
9

Mark (√) against the correct answer in each of the following:

The angle between the vectors and is
10

Mark (√) against the correct answer in each of the following:

If and then the angle between and is
10

Mark (√) against the correct answer in each of the following:

If and then the angle between and is
11

Mark (√) against the correct answer in each of the following:

If and then the angle between and is
11

Mark (√) against the correct answer in each of the following:

If and then the angle between and is
12

Mark (√) against the correct answer in each of the following:

If and be such that then λ = ?
12

Mark (√) against the correct answer in each of the following:

If and be such that then λ = ?
13

Mark (√) against the correct answer in each of the following:

What is the projection of on ?
13

Mark (√) against the correct answer in each of the following:

What is the projection of on ?
14

Mark (√) against the correct answer in each of the following:

If then
14

Mark (√) against the correct answer in each of the following:

If then
15

Mark (√) against the correct answer in each of the following:

If and are mutually perpendicular unit vectors then


15

Mark (√) against the correct answer in each of the following:

If and are mutually perpendicular unit vectors then


16

Mark (√) against the correct answer in each of the following:

If the vectors and are perpendicular to each other then λ = ?
16

Mark (√) against the correct answer in each of the following:

If the vectors and are perpendicular to each other then λ = ?
17

Mark (√) against the correct answer in each of the following:

If θ is the angle between two unit vectors and then


17

Mark (√) against the correct answer in each of the following:

If θ is the angle between two unit vectors and then


18

Mark (√) against the correct answer in each of the following:

If and then


18

Mark (√) against the correct answer in each of the following:

If and then


19

Mark (√) against the correct answer in each of the following:

If and then


19

Mark (√) against the correct answer in each of the following:

If and then


20

Mark (√) against the correct answer in each of the following:

If and then the angle between and is
20

Mark (√) against the correct answer in each of the following:

If and then the angle between and is
21

Mark (√) against the correct answer in each of the following:

If and then


21

Mark (√) against the correct answer in each of the following:

If and then


22

Mark (√) against the correct answer in each of the following:

Two adjacent sides of a || gm are represented by the vectors and The area of the || gm is
22

Mark (√) against the correct answer in each of the following:

Two adjacent sides of a || gm are represented by the vectors and The area of the || gm is
23

Mark (√) against the correct answer in each of the following:

The diagonals of a || gm are represented by the vectors and The area of the || gm is
23

Mark (√) against the correct answer in each of the following:

The diagonals of a || gm are represented by the vectors and The area of the || gm is
1

Mark (√) against the correct answer in each of the following:

Two adjacent sides of a triangle are represented by the vectors and The area of the triangle is
1

Mark (√) against the correct answer in each of the following:

Two adjacent sides of a triangle are represented by the vectors and The area of the triangle is
25

Mark (√) against the correct answer in each of the following:

The unit vector normal to the plane containing and is
25

Mark (√) against the correct answer in each of the following:

The unit vector normal to the plane containing and is
26

Mark (√) against the correct answer in each of the following:

If and are unit vectors such that then


26

Mark (√) against the correct answer in each of the following:

If and are unit vectors such that then


27

Mark (√) against the correct answer in each of the following:

If and are mutually perpendicular unit vectors then


27

Mark (√) against the correct answer in each of the following:

If and are mutually perpendicular unit vectors then


28

Mark (√) against the correct answer in each of the following:
28

Mark (√) against the correct answer in each of the following:
29

Mark (√) against the correct answer in each of the following:

If and be the coterminous edges of a parallelepiped then its volume is
29

Mark (√) against the correct answer in each of the following:

If and be the coterminous edges of a parallelepiped then its volume is
30

Mark (√) against the correct answer in each of the following:

If the volume of a parallelepiped having and as conterminous edges, is 216 cubic units then the value of λ is
30

Mark (√) against the correct answer in each of the following:

If the volume of a parallelepiped having and as conterminous edges, is 216 cubic units then the value of λ is
31

Mark (√) against the correct answer in each of the following:

It is given that the vectors and are coplanar. Then, the value of λ is
31

Mark (√) against the correct answer in each of the following:

It is given that the vectors and are coplanar. Then, the value of λ is
32

Mark (√) against the correct answer in each of the following:

Which of the following is meaningless?
32

Mark (√) against the correct answer in each of the following:

Which of the following is meaningless?
33

Mark (√) against the correct answer in each of the following:
33

Mark (√) against the correct answer in each of the following:
34

Mark (√) against the correct answer in each of the following:

For any three vectors the value of is
34

Mark (√) against the correct answer in each of the following:

For any three vectors the value of is
1

Mark (√) against the correct answer in the following:

A unit vector in the direction of the vector is
2

Mark (√) against the correct answer in the following:

The direction cosines of the vector are
3

Mark (√) against the correct answer in the following:

If A(1, 2, -3) and B(-1, -2, 1) are the end points of a vector then the direction cosines of are
4

Mark (√) against the correct answer in the following:

If a vector makes angle α, β and γ with the x-axis, y-axis and z-axis respectively then the value of (sin2α + sin2β + sin2γ) is
5

Mark (√) against the correct answer in the following:

The vector is a
6

Mark (√) against the correct answer in the following:

What is the angle which the vector makes with the z-axis?
7

Mark (√) against the correct answer in the following:

if and are vectors such that and then the angle between and is
8

Mark (√) against the correct answer in the following:

If and are two vectors such that and then the angle between and is
9

Mark (√) against the correct answer in the following:

The angle between the vectors and is
10

Mark (√) against the correct answer in the following:

If and then the angle between and is
11

Mark (√) against the correct answer in the following:

If and then the angle between and is
12

Mark (√) against the correct answer in the following:

If and be such that then λ = ?
13

Mark (√) against the correct answer in the following:

What is the projection of on ?
14

Mark (√) against the correct answer in the following:

If then
15

Mark (√) against the correct answer in the following:

If and are mutually perpendicular unit vectors then


16

Mark (√) against the correct answer in the following:

If and are mutually perpendicular unit vectors then


17

Mark (√) against the correct answer in the following:

If θ is the angle between two unit vectors and then


18

Mark (√) against the correct answer in the following:

If and then


19

Mark (√) against the correct answer in the following:

If and then


20

Mark (√) against the correct answer in the following:

If and then the angle between and is
21

Mark (√) against the correct answer in the following:

If and then


22

Mark (√) against the correct answer in the following:

Two adjacent sides of a || gm are represented by the vectors and The area of the || gm is
23

Mark (√) against the correct answer in the following:

The diagonals of a || gm are represented by the vectors and The area of the || gm is
24

Mark (√) against the correct answer in the following:

Two adjacent sides of a triangle are represented by the vectors and The area of the triangle is
25

Mark (√) against the correct answer in the following:

The unit vector normal to the plane containing and is
26

Mark (√) against the correct answer in the following:

If and are unit vectors such that then


27

Mark (√) against the correct answer in the following:

If and are mutually perpendicular unit vectors then


28

Mark (√) against the correct answer in the following:
29

Mark (√) against the correct answer in the following:

If and be the coterminous edges of a parallelepiped then its volume is
30

Mark (√) against the correct answer in the following:

If the volume of a parallelepiped having and as conterminous edges, is 216 cubic units then the value of λ is
31

Mark (√) against the correct answer in the following:

It is given that the vectors and are coplanar. Then, the value of λ is
32

Mark (√) against the correct answer in the following:

Which of the following is meaningless?
33

Mark (√) against the correct answer in the following:
34

Mark (√) against the correct answer in the following:

For any three vectors the value of is