# MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra

## MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra

### Vector Algebra Important Questions

Vector Algebra Objective Type Questions

Question 1.

Question 1.
Unit vector parallel to the resultant vector of vectors 2$$\hat { i }$$ + 4$$\hat { j }$$ – 5$$\hat { k }$$ and $$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$ is:
(a) – $$\hat { i }$$ – $$\hat { j }$$ + 8$$\hat { k }$$
(b) $$\frac { 3\hat { i } +6\hat { j } -2\hat { k } \quad }{ 7 }$$
(c) $$\frac { -\hat { i } -+8\hat { k } \quad }{ \sqrt { 69 } }$$
(d) $$\frac { -\hat { i } +2\hat { j } -8\hat { k } \quad }{ \sqrt { 69 } }$$

Question 2.
If $$\vec { O }$$A = a, $$\vec { O }$$B = b and C is a point on AB such that $$\vec { A }$$C = 3AB, then $$\vec { O }$$C is equal to:
(a) 3$$\vec { a }$$ – 2$$\vec { b }$$
(b) 3$$\vec { b }$$ – 2$$\vec { a }$$
(c) 3$$\vec { a }$$ – $$\vec { b }$$
(d) 3$$\vec { b }$$ – $$\vec { a }$$

Question 3.
If $$\vec { a }$$ and $$\vec { b }$$ are two vectors such that |$$\vec { a }$$| = 2, |$$\vec { b }$$| = 1 and $$\vec { a }$$.$$\vec { b }$$ = $$\sqrt { 3 }$$, then the angle between them is:
(a) $$\frac { \pi }{ 2 }$$
(b) $$\frac { \pi }{ 4 }$$
(c) $$\frac { \pi }{ 6 }$$
(d) $$\frac { \pi }{ 7 }$$

Question 4.
Area of parallelogram whose adjacent sides are $$\hat { i }$$ – 2$$\hat { j }$$ + 3$$\hat { k }$$ and 2$$\hat { i }$$ + $$\hat { j }$$ – 4$$\hat { k }$$ is:
(a) 3$$\sqrt{6}$$
(b) 4$$\sqrt{6}$$
(c) 5$$\sqrt{6}$$
(d) 6$$\sqrt{6}$$

Question 5.
If $$\vec { a }$$ = $$\vec { b }$$ + $$\vec { c }$$, then $$\vec { a }$$.( $$\vec { b }$$ × $$\vec { c }$$ ) is equal to:
(a) 2$$\vec { a }$$. ( $$\vec { b }$$ + $$\vec { c }$$ )
(b) 0
(c) $$\vec { b }$$ = ( $$\vec { a }$$ + $$\vec { c }$$ )
(d) None of these

Question 2.
Fill in the blanks:

1. Sum or difference of two vectors is always a ………………………….
2. Addition of vectors obeys ………………………….
3. ( $$\vec { a }$$ + $$\vec { b }$$ ) + $$\vec { c }$$ = $$\vec { a }$$ + …………………………..
4. Addition of two vectors can be obtained from ……………………………..
5. Position vector of point (1,2, 3) w.r.t. the origin will be ……………………………..
6. If $$\vec { a }$$ and $$\vec { b }$$ are parllel then $$\vec { a }$$ × $$\vec { b }$$ = …………………………..
7. If $$\vec { a }$$ and $$\vec { b }$$ are parallel then $$\vec { a }$$ × $$\vec { a }$$ = ………………………..
8. The unit vector in the direction of vector $$\vec { a }$$ will be ……………………………
9. The projection of $$\vec { b }$$ along the direction of $$\vec { a }$$ will be ……………………………..
10. If the vectors 2$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$ 3$$\hat { i }$$ + p$$\hat { j }$$ + 5$$\hat { k }$$ are coplanar then value of p will be …………………………….
11. A force 2$$\hat { i }$$ + $$\hat { j }$$ + $$\hat { k }$$, acts at a point A whose position vector 2$$\hat { i }$$ – $$\hat { j }$$ The moment of the force with respect to the origin will be ………………………………..
12. The area of the parallelogram will be …………………………. whose diagonals are 3$$\hat { i }$$ + $$\hat { j }$$ – 2$$\hat { k }$$ and $$\hat { i }$$ – 3$$\hat { j }$$ + 4$$\hat { k }$$.

1. New vector
2. Commutative and associative law
3. ( $$\vec { b }$$ + $$\vec { c }$$ )
4. Traingle law of vector addition
5. $$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$
6. collinear
7. $$\vec { O }$$
8. $$\frac { \vec { a } }{ |\vec { a } | }$$
9. $$\frac { \vec { a } .\vec { b } }{ |\vec { a } | }$$
10. -4
11. $$\hat { i }$$ + 2$$\hat { j }$$ + 4$$\hat { k }$$
12. 5$$\sqrt { 3 }$$ sq. unit.

Question 3.
Write True/False:

1. The sum of the vectors determined by the sides of a triangle taken in order is zero.
2. If $$\vec { a }$$ and $$\vec { b }$$ are two non collinear vectors, then |$$\vec { a }$$ + $$\vec { b }$$| ≥ |$$\vec { a }$$ + $$\vec { b }$$|
3. A vector whose initial and terminal points are coincident is called unit vector.
4. If the position vector of the points P and Q are $$\hat { i }$$ + 3$$\hat { j }$$ – 7$$\hat { k }$$ and 5$$\hat { i }$$ – 2$$\hat { j }$$ + 4$$\hat { k }$$ respectively, then the value of |$$\vec { P }$$Q| is 9$$\sqrt { 2 }$$.
5. If |$$\vec { a }$$ + $$\vec { a }$$|=|$$\vec { a }$$ – $$\vec { b }$$|, then $$\vec { a }$$ × $$\vec { b }$$ = $$\vec { 0 }$$.
6. The value of $$\vec { a }$$.( $$\vec { a }$$ × $$\vec { b }$$ ) is zero.
7. If the vectors $$\hat { i }$$ – λ$$\hat { j }$$ + $$\hat { k }$$ and $$\hat { i }$$ – $$\hat { j }$$ + 5$$\hat { k }$$ are mutually perpendicular, then the value of λ is 6.

1. True
2. False
3. False
4. True
5. False
6. True
7. False.

Question 4.
Write the answer is one word/sentence:

1. If $$\vec { a }$$, $$\vec { b }$$, $$\vec { c }$$ are the position vectors of the vectors of the ∆ABC, then write the formula for area of ∆ABC.
2. If $$\vec { a }$$ = $$\hat { i }$$ – 2$$\hat { j }$$ + 3$$\hat { k }$$,$$\vec { b }$$ = 2$$\hat { i }$$ + $$\hat { j }$$ – $$\hat { k }$$ and $$\vec { c }$$ = $$\hat { j }$$ + $$\hat { k }$$, then find the value of [ $$\vec { a }$$ $$\vec { b }$$ $$\vec { c }$$ ]
3. Find the angle between two vector 3$$\hat { i }$$ – 2$$\hat { j }$$ + 4$$\hat { k }$$ and $$\hat { i }$$ – $$\hat { j }$$ + 5$$\hat { k }$$.
4. Find the value of $$\hat { i }$$ × ( $$\hat { j }$$ + 3$$\hat { k }$$ ) + $$\hat { j }$$ × ( $$\hat { k }$$ + $$\hat { i }$$ ) + $$\hat { k }$$ × ( $$\hat { i }$$ + $$\hat { j }$$ )
5. Find the projection of $$\vec { a }$$ in the direction of $$\vec { b }$$.
6. If $$\vec { a }$$ and $$\vec { b }$$ are mutually perpendicular vector then find, the value ( $$\vec { a }$$ + $$\vec { b }$$ ) 2

1. $$\frac{1}{2}$$ |$$\vec { a }$$ × $$\vec { b }$$ + $$\vec { b }$$ × $$\vec { c }$$ + $$\vec { c }$$ × $$\vec { a }$$|
2. 12
3. cos-1 $$\frac { 25 }{ \sqrt { 783 } }$$
4. 0
5. $$\frac { \vec { a } .\vec { b } }{ |\vec { b } | }$$
6. |$$\vec { a }$$|2 + |$$\vec { b }$$|2

Question 4.
Match the Column:

1. (d)
2. (e)
3. (a)
4. (b)
5. (c)
6. (g)
7. (f).

Vector Algebra Very Short Answer Type Questions

Question 1.
Given vectors $$\vec { a }$$ = $$\hat { i }$$ – 2$$\hat { j }$$ + $$\hat { k }$$, $$\vec { b }$$ = – 2$$\hat { i }$$ + 4$$\hat { j }$$ + 5$$\hat { k }$$ and $$\vec { c }$$ = $$\hat { i }$$ – 6$$\hat { j }$$ – 7$$\hat { k }$$. Then find the value of |$$\vec { a }$$ + $$\vec { b }$$ + $$\vec { c }$$|? (NCERT, CBSE 2012)
Solution:

Question 2.
Find the unit vector in the direction of sum of the vectors $$\vec { a }$$ = 2$$\hat { i }$$ – $$\hat { j }$$ + 2$$\hat { k }$$ and $$\vec { b }$$ = – $$\hat { i }$$ + $$\hat { j }$$ + 3$$\hat { k }$$?
Solution:

Question 3.
Find the vector in the direction of vector $$\vec { a }$$ = $$\hat { i }$$ – 2$$\hat { j }$$ which has magnitude 7 units? (NCERT)
Solution:
$$\vec { a }$$ = $$\hat { i }$$ – 2$$\hat { j }$$
Unit vector in the direction of given vector a is:

The vector having magnitude be equal to 7:

Question 4.
Prove that the vectors 2$$\hat { i }$$ – 3$$\hat { j }$$ + 4$$\hat { k }$$ and -4$$\hat { i }$$ + 6$$\hat { j }$$ – 8$$\hat { k }$$ are collinear? (NCERT)
Solution:

Hence vector $$\vec { a }$$, $$\vec { b }$$ are collinear. Proved.

Question 5.
Find direction cosine of the vector $$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$? (NCERT)
Solution:

Question 6.
If $$\vec { a }$$ = 2 $$\hat { i }$$ – 3 $$\hat { j }$$ + $$\hat { k }$$ and $$\vec { a }$$ = $$\hat { i }$$ + $$\hat { j }$$ – 2$$\hat { k }$$, then find $$\vec { a }$$ – $$\vec { b }$$?
Solution:

Question 7.
If $$\vec { a }$$ = $$\hat { i }$$ + $$\hat { j }$$ + 2$$\hat { k }$$ and $$\vec { b }$$ = 3$$\hat { i }$$ + 2$$\hat { j }$$ – $$\hat { k }$$, then find |2$$\vec { a }$$ – $$\vec { b }$$|?
Solution:

Question 8.
If the vector $$\vec { a }$$ = 2$$\hat { i }$$ + $$\hat { j }$$ + $$\hat { k }$$ and $$\vec { b }$$ = $$\hat { i }$$ – 4$$\hat { j }$$ + λ$$\hat { k }$$ are perpendicular then find the value of λ?
Solution:
The given vectors are perpendicular
Hence $$\vec { a }$$ . $$\vec { b }$$ = 0
(2$$\hat { i }$$ + $$\hat { j }$$ + $$\hat { k }$$ ). ( $$\hat { i }$$ – 4$$\hat { j }$$ + λ$$\hat { k }$$ ) = 0
⇒ 2 – 4 + λ = 0
⇒ λ = 2.

Question 9.
(A) Prove that the vectors 2$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$ and –$$\hat { i }$$ + 3$$\hat { j }$$ + 5$$\hat { k }$$ are perpendicular to each other?
Solution:

L.H.S = – 2 – 3 + 5
= 0 = R.H.S. Proved

(B) Prove that vector 3$$\hat { i }$$ – 2$$\hat { j }$$ + $$\hat { k }$$ and 2$$\hat { i }$$ + $$\hat { j }$$ – 4$$\hat { k }$$ are perpendicular?
Solution:
Solve as Q.No. 9(A)

Question 10.
If $$\vec { a }$$ = 4$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$ and $$\vec { b }$$ p$$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$ are perpendicular. Find the value of p?
Solution:
$$\vec { a }$$ = 4$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$ and $$\vec { b }$$ p$$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$
$$\vec { a }$$ and $$\vec { b }$$ are perpendicular
$$\vec { a }$$.$$\vec { b }$$ = 0
∴ 4p – 2 + 3 = 0
⇒ 4p = -1
⇑ p = – $$\frac{1}{4}$$

Question 11.
(A) Find the angle between the vectors (2$$\hat { i }$$ + 3$$\hat { j }$$ – 4$$\hat { k }$$ ) and (3$$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$ )?
Solution:
Let $$\vec { a }$$ = 2$$\hat { i }$$ + 3$$\hat { j }$$ – 4$$\hat { k }$$, $$\vec { b }$$ = 3$$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$
Let θ be the angle between them

(B) Find the angle between vectors $$\vec { a }$$ = 2$$\hat { i }$$ – 2$$\hat { j }$$ – $$\hat { k }$$ and $$\vec { b }$$ = 6$$\hat { i }$$ – 3$$\hat { j }$$ + 2$$\hat { k }$$?
Solution:
Solve as Q.No. 11(A)

(C) If $$\vec { a }$$ = 2$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$ and $$\vec { b }$$ = 3$$\hat { i }$$ – 4$$\hat { j }$$ – 4$$\hat { k }$$, then find their dot product and angle between them?
Solution:

If θ be the angle between them

Question 12.
If |$$\bar { |a| }$$ = 10, $$\bar { |b| }$$ = 2 and $$\bar { a }$$. $$\bar { b }$$ = 2 and $$\bar { a }$$. $$\bar { b }$$ = 12, then find the value of |$$\bar { a }$$ × $$\bar { b }$$?
Solution:

Question 13.
If $$\vec { a }$$ = $$\hat { i }$$ + $$\hat { j }$$ + $$\hat { k }$$ and $$\vec { b }$$ = $$\hat { i }$$ – $$\hat { j }$$ – $$\hat { k }$$, then find $$\vec { a }$$ × $$\vec { b }$$?
Solution:

Question 14.
A force $$\vec { F }$$ = 4$$\hat { i }$$ – 3$$\hat { j }$$ + 2$$\hat { k }$$ is acting along the direction $$\vec { d }$$ = – $$\hat { i }$$ – 3$$\hat { j }$$ + 5$$\hat { k }$$? Find the work done by the force?
Solution:
$$\vec { d }$$ = – $$\hat { i }$$ – 3$$\hat { j }$$ + 5$$\hat { k }$$, $$\vec { F }$$ = 4$$\hat { i }$$ – 3$$\hat { j }$$ + 2$$\hat { k }$$ (given)
∴ Work done by force
W = $$\vec { F }$$. $$\vec { d }$$
= (4$$\hat { i }$$ – 3$$\hat { j }$$ + 2$$\hat { k }$$ ). (-$$\hat { i }$$ – 3$$\hat { j }$$ + 5$$\hat { k }$$ )
= -4 + 9 + 10 = 15 unit.

Question 15.
If |$$\vec { a }$$ + $$\vec { b }$$| = |$$\vec { a }$$ – $$\vec { b }$$|, then prove that $$\vec { a }$$ and $$\vec { b }$$ are perpendicular?
Solution:

Since dot product is zero. So vectors $$\vec { a }$$ and $$\vec { b }$$ are perpendiculars. Proved.

Question 16.
If $$\vec { a }$$ and $$\vec { b }$$ are two vectors such that |$$\vec { a }$$| = 2, |$$\vec { b }$$| = 3 and $$\vec { a }$$. $$\vec { a }$$ = 3, then find angle between $$\vec { a }$$ and $$\vec { b }$$?
Solution:
Solve as Q.No. 17

Question 17.
If |$$\vec { a }$$| = 4, |$$\vec { b }$$| = 4 and $$\vec { a }$$. $$\vec { b }$$ = 6, then find the angle between $$\vec { a }$$ and $$\vec { b }$$?
Solution:

Question 18.
If $$\vec { a }$$ and $$\vec { b }$$ are two two vectors such that |$$\vec { a }$$| = 2, |$$\vec { b }$$| = 7 and $$\vec { a }$$ ×
$$\vec { b }$$ = 3$$\hat { i }$$ + 2$$\hat { j }$$ + 6$$\hat { k }$$, then find the angle between $$\vec { a }$$ and $$\vec { b }$$?
Solution:
Let θ be the angle between $$\vec { a }$$ and $$\vec { b }$$

Question 19.
Find cosine angle between vectors 2$$\hat { i }$$ – 3$$\hat { j }$$ + $$\hat { k }$$ and $$\hat { i }$$ + $$\hat { j }$$ – 2$$\hat { k }$$?
Solution:

Question 20.
Find the area of the parallelogram whose two adjacent sides are represented by the vectors $$\vec { a }$$ = 2$$\hat { i }$$ – 3$$\hat { j }$$ + $$\hat { k }$$, $$\vec { b }$$ = $$\hat { i }$$ – $$\hat { j }$$ + 2$$\hat { k }$$ and $$\vec { c }$$ = 2$$\hat { i }$$ + $$\hat { j }$$ – $$\hat { k }$$?
Solution:

= 2(1 – 2) + 3(-1-4) + 1(1 + 2)
= -2 – 15 + 3
= -14 cubic unit

Question 21.
Prove that:
$$\hat { i }$$.( $$\hat { j }$$ × $$\hat { k }$$ + ( $$\hat { i }$$ × $$\hat { k }$$). $$\hat { j }$$ = 0?
Solution:

Question 22.
If vectors $$\vec { a }$$ and $$\vec { b }$$ are perpendicular then prove that |$$\vec { a }$$ + $$\vec { b }$$|2 = |$$\vec { a }$$|2 + |$$\vec { b }$$|2?
Solution:
We know that
|$$\vec { a }$$ + $$\vec { b }$$|2 = |$$\vec { a }$$|2 + |$$\vec { b }$$|2
Vector $$\vec { a }$$ and $$\vec { b }$$ are perpendicular, then
$$\vec { a }$$. $$\vec { b }$$ = 0
⇒ |$$\vec { a }$$ + $$\vec { a }$$|2 + |$$\vec { a }$$|2 + |$$\vec { b }$$|2. Proved.

Question 23.
Prove that:
$$\vec { a }$$ × ( $$\vec { b }$$ + $$\vec { c }$$ ) + $$\vec { b }$$ × ( $$\vec { c }$$ + $$\vec { a }$$ ) + $$\vec { c }$$ × ( $$\vec { a }$$ + $$\vec { b }$$ ) = $$\vec { 0 }$$?
Solution:

Question 24.
Find the work done by the force $$\vec { F }$$ = 2$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$ in the direction $$\vec { d }$$ = 3$$\hat { i }$$ + 2$$\hat { j }$$ + 5$$\hat { k }$$?
Solution:
W = $$\vec { F }$$. $$\vec { d }$$
= (2$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$ ). (3$$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$ )
= 6 – 2 + 3 = 7 unit.

Question 25.
If modulus of two vectors $$\vec { a }$$ and $$\vec { a }$$ are equal and angle between them is 60° and their dot product is $$\frac{9}{2}$$ find their modulus? (CBSE 2018)
Solution:

Question 26.
Find the area of the parallelogram whose adjacent sides are given by vectors $$\vec { a }$$ = 2$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$ and $$\vec { b }$$ = 3$$\hat { i }$$ + 4$$\hat { j }$$ – $$\hat { k }$$?
Solution:

Question 27.
If $$\vec { a }$$ = 4$$\hat { i }$$ + $$\hat { j }$$ + $$\hat { k }$$, $$\vec { b }$$ = $$\hat { i }$$ – 2$$\hat { k }$$ then find the value of |2$$\vec { b }$$ × $$\vec { a }$$|?
Solution:

Question 28.
If $$\vec { a }$$ = 4$$\hat { i }$$ + 3$$\hat { j }$$ + 3$$\hat { k }$$ and $$\vec { b }$$ = 3$$\hat { i }$$ + 2$$\hat { k }$$ then, find the value of |$$\vec { b }$$ × 2$$\vec { a }$$|?
Solution:

Question 29.
If $$\vec { a }$$ = 2$$\hat { i }$$ + $$\hat { j }$$ + 2$$\hat { k }$$ and $$\vec { b }$$ = 5$$\hat { i }$$ – 3$$\hat { j }$$ + $$\hat { k }$$, then find the magnitude of vector $$\vec { b }$$ in the direction of $$\vec { a }$$?
Solution:

Question 30.
If $$\vec { a }$$ = $$\hat { i }$$ + 3$$\hat { j }$$ – 2$$\hat { k }$$, $$\vec { b }$$ = – $$\hat { j }$$ + 3$$\hat { k }$$ then find the value |$$\vec { a }$$ × $$\vec { b }$$|?
Solution:

Vector Algebra Short Answer Type Questions

Question 1.
Prove that: A(-2$$\hat { i }$$ + 3$$\hat { j }$$ + 5$$\hat { k }$$ ), B( $$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$ ) and C(7$$\hat { i }$$ + 0$$\hat { j }$$ – $$\hat { k }$$ ) are coplanar? (NCERT)
Solution:
Let O be the origin then position vector of A, B and C is

Hence vector $$\vec { A }$$B and $$\vec { B }$$C are parallel but $$\vec { A }$$B and $$\vec { B }$$C has common point B. Hence points A, B and C are coplanar.

Question 2.
If position vectors of points A, B, C and D are 2$$\hat { i }$$ + 4$$\hat { k }$$, 5$$\hat { i }$$ + 3$$\sqrt { 3 }$$ $$\hat { j }$$ + 4$$\hat { k }$$, -2$$\sqrt { 3 }$$ $$\hat { j }$$ + $$\hat { k }$$ then prove that:
CD||AB and CD = $$\frac{2}{3}$$ $$\vec { A }$$B?
Solution:
Let O be the origin

Question 3.
If G is centroid of ∆ABC, then prove that:
$$\vec { G }$$A + $$\vec { G }$$B + $$\vec { G }$$C = $$\vec { 0 }$$?
Solution:
Let vectors of vertices A,B and C of ∆ABC are $$\vec { a }$$, $$\vec { b }$$ and $$\vec { c }$$ respectively.
∴ Position vector of centroid G = $$\frac { \vec { a } +\vec { b } +\vec { c } }{ 3 }$$

Question 4.
Using vectors prove that the medians of traiangle are concurrent?
Solution:
Let medium of ∆ABC are AD, BE and CF.
Let $$\vec { a }$$, $$\vec { b }$$ and $$\vec { c }$$ be the positive vector of points A, B and C respectively.

Now position vector of a point dividing the median AD in the ratio 2 : 1 is

Position vector of a point which divides median BE in the ratio of 2 : 1 is

Position vector of a point which divides median BE in the ratio of 2 : 1 is

Hence, medians of triangle meets at point G it means concurrent whose position vector is $$\frac { \vec { a } +\vec { b } +\vec { c } }{ 3 }$$. Point G is centroid of traingle.

Question 5.
A vector $$\vec { O }$$P, makes angle 45° with OX and 60° with OY. Find the angle made by $$\vec { O }$$P with OZ?
Solution:
Let angle made by vector $$\vec { O }$$P with axes OX, OY and OZ are α, β, γ respectively. then
α = 45°,
β = 60°
∴ l = cos α = cos 45° = $$\frac { 1 }{ \sqrt { 2 } }$$
m = cos β = cos 60° = $$\frac{1}{2}$$
and n = cos γ
We know that
l2 + m2 + n2 = 1

Question 6.
Find the vector $$\vec { a }$$ which makes an angle with X – axis, F – axis and Z – axis respectively are $$\frac { \pi }{ 4 }$$, $$\frac { \pi }{ 2 }$$ and angle θ and its magnitude is 5$$\sqrt { 2 }$$?
Solution:
Given:
α = $$\frac { \pi }{ 4 }$$,
β = $$\frac { \pi }{ 2 }$$, γ = θ
∴l = cos $$\frac { \pi }{ 4 }$$ = $$\frac { 1 }{ \sqrt { 2 } }$$, m = cos $$\frac { \pi }{ 2 }$$ = 0, n = cos θ.
We know that

Direction cosine of vector $$\frac { 1 }{ \sqrt { 2 } }$$, 0 , $$\frac { 1 }{ \sqrt { 2 } }$$

Question 7.
Prove that:
( $$\vec { a }$$ × $$\vec { b }$$ )2 = a2b2 – ( $$\vec { a }$$.$$\vec { b }$$ )2?
Solution:
L.H.S = ( ( $$\vec { a }$$ × $$\vec { b }$$ )2 = ( $$\vec { a }$$ × $$\vec { b }$$ ).( $$\vec { a }$$ ×  $$\vec { b }$$ )
= (ab sin θ$$\hat { n }$$ ). (ab sin θ $$\hat { n }$$ ) = a2 b2 sin2θ,
= a2 b2 (1 – cos2θ)
= a2 b2 – a2 b2cos2θ
= a2 b2 – (ab cos θ)2
= a2 b2 – ( $$\vec { a }$$. $$\vec { b }$$ )2 = R.H.S Proved.

Question 8.
If $$\vec { a }$$ = 2$$\hat { i }$$ – 3$$\hat { j }$$ + $$\hat { k }$$, $$\vec { b }$$ = $$\hat { i }$$ – $$\hat { j }$$ + 2$$\hat { k }$$ and $$\vec { c }$$ = 2$$\hat { i }$$ + $$\hat { j }$$ – $$\hat { k }$$ then find the value of $$\vec { a }$$ × ( $$\vec { b }$$ × $$\vec { c }$$ )?
Solution:

Question 9.
Find the volume of parallel cuboid whose vectors of three faces are denoted by: $$\hat { i }$$ + $$\hat { j }$$ + $$\hat { k }$$, $$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$, $$\hat { i }$$ + $$\hat { j }$$ – $$\hat { k }$$?
Solution:

Question 10.
If $$\vec { a }$$ = 2$$\hat { i }$$ – 3$$\hat { j }$$ + $$\hat { k }$$, $$\vec { b }$$ = $$\hat { i }$$ – $$\hat { j }$$ + 2$$\hat { k }$$ and $$\vec { c }$$ = 2$$\hat { i }$$ + $$\hat { j }$$ – $$\hat { k }$$ then find the value of [ $$\vec { a }$$ $$\vec { b }$$ $$\vec { c }$$ ]
Solution:

Question 11.
If $$\vec { a }$$ = 3$$\hat { i }$$ – $$\hat { j }$$ + 2$$\hat { k }$$, $$\vec { b }$$ = 2$$\hat { i }$$ + $$\hat { j }$$ – $$\hat { k }$$ and $$\vec { c }$$ = $$\hat { i }$$ – 2$$\hat { j }$$ + 2$$\hat { k }$$ then, find the value of $$\vec { a }$$, $$\vec { b }$$, $$\vec { c }$$?
Solution:
Solve like Q.No.10.

Question 12.
If $$\vec { a }$$ = $$\hat { i }$$ – 2$$\hat { j }$$ + 3$$\hat { k }$$, $$\vec { b }$$ = – $$\hat { i }$$ + 3 $$\hat { j }$$ – 4 $$\hat { k }$$ and $$\vec { c }$$ = $$\hat { i }$$ – 3$$\hat { j }$$ + 5$$\hat { k }$$ then prove that $$\vec { a }$$, $$\vec { b }$$, $$\vec { c }$$ are coplanar?
Solution:
If $$\vec { a }$$, $$\vec { b }$$, $$\vec { c }$$ are coplanar then

Question 13.
Prove that 2$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$, $$\hat { i }$$ + 2$$\hat { j }$$ – 3$$\hat { k }$$ and 3$$\hat { i }$$ – 4$$\hat { j }$$ + 5k are coplanar?
Solution:
Let $$\vec { a }$$ = 2$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$, $$\hat { i }$$ + 2$$\hat { j }$$ – 3$$\hat { k }$$ and 3$$\hat { i }$$ – 4$$\hat { j }$$ + 5$$\hat { k }$$

Question 14.
(A) Find the value of λ for which the vectors λ$$\hat { i }$$ + 3$$\hat { j }$$ + 2$$\hat { k }$$, 2$$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$ and 2$$\hat { i }$$ + 3$$\hat { j }$$ + 4$$\hat { k }$$ are coplanar?
Solution:
Let $$\vec { a }$$ = λ$$\hat { i }$$ + 3$$\hat { j }$$ + 2$$\hat { k }$$, $$\vec { b }$$ = 2$$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$ and 2$$\hat { i }$$ + 3$$\hat { j }$$ + 4$$\hat { k }$$ are coplanar?
Solution:
Let $$\vec { a }$$ = λ$$\hat { i }$$ + 3$$\hat { j }$$ + 2$$\hat { k }$$, $$\vec { b }$$ = 2$$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$, $$\vec { c }$$ = 2$$\hat { i }$$ + 3$$\hat { j }$$ + 4$$\hat { k }$$
Given vector are coplanar if
[ $$\vec { a }$$ $$\vec { b }$$ $$\vec { c }$$ ] = 0
$$\left|\begin{array}{lll} {\lambda} & {3} & {2} \\ {2} & {2} & {3} \\ {2} & {3} & {4} \end{array}\right|$$ = 0
⇒ λ(8 – 9) -2(12 – 6+ 2 (9 – 4) = 0
⇒ -λ – 12 + 10 = 0
⇒ λ = -2.

(B) Find the value of λ for which given vectors are coplanar
$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$, 2$$\hat { i }$$ + $$\hat { j }$$ – $$\hat { k }$$, λ$$\hat { i }$$ – $$\hat { j }$$ + λ$$\hat { k }$$
Solution:
Solve like Q.No. 14 (A)
λ = 1

(C) Find the value of λ for which the given vectors are coplanar 2$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$, $$\hat { i }$$ + 2$$\hat { j }$$ – 3$$\hat { k }$$ and 3$$\hat { i }$$ + λ$$\hat { j }$$ + 5$$\hat { k }$$?
Solution:
Solve like Q.No. 14 (A)
λ = – $$\frac{18}{5}$$

Question 15.
If the angle between two unit vectors $$\vec { a }$$ and $$\vec { b }$$ is θ then prove that:
cos $$\frac { \theta }{ 2 }$$ = $$\frac{1}{2}$$ |$$\bar { a }$$ + $$\bar { b }$$| is θ then prove that:
sin $$\frac { \theta }{ 2 }$$ = $$\frac{1}{2}$$ |$$\bar { a }$$ – $$\bar { b }$$|
Solution:

Question 16.
The angle between two vectors $$\vec { a }$$ and $$\vec { b }$$ is θ then prove that:
sin $$\frac { \theta }{ 2 }$$ = $$\frac{1}{2}$$ |$$\bar { a }$$ – $$\bar { b }$$|
Solution:
sin $$\frac { \theta }{ 2 }$$ = $$\frac{1}{2}$$ |$$\bar { a }$$ – $$\bar { b }$$|

Question 17.
In any traiangle prove that ABC?
(A) ac cos B – bc cos A = a2 – b2?
(B) 2(bc cos A + ca cos B + ab cos C) = a2 + b2 + c2?
Solution:

Question 18.
In ∆ABC prove by vector method c = acosB + bcosA?
Solution:
In ∆ABC

⇒ c2 = ac cos B + bc cos A
⇒ c2 = c(a cos B + b cos A)
⇒ c = a cos B + b cos A. Proved.

Question 19.
In ∆ABC prove by vector method
b2 = a2 + c2 – 2ac cos B?
Solution:
In ∆ABC we know that

Question 20.
In ∆ABC Prove the following:
(A) a2 = b2 + c2 – 2bc cos A?
(B) c2 = a2 + b2 – 2ab cos C?
Solution:
Solve like Q.No. 19

Question 21.
(A) Find the unit vector normal to the vector $$\vec { a }$$ = 2$$\hat { i }$$ + 2$$\hat { j }$$ + $$\hat { k }$$ and $$\vec { b }$$
= 4$$\hat { i }$$ + 4$$\hat { j }$$ – 7$$\hat { k }$$?
Solution:

(B) Find the unit vector normal to the vectors $$\vec { a }$$ = $$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$ and $$\vec { b }$$ =
$$\hat { i }$$ + 2$$\hat { j }$$ – $$\hat { k }$$?
Solution:
Solve like Q.No. 21 (A)

(C) Find the unit vector normal to the vectors $$\vec { a }$$ = 3$$\hat { i }$$ + $$\hat { j }$$ – 2$$\hat { k }$$ and $$\vec { b }$$ = 2$$\hat { i }$$ + 3$$\hat { j }$$ – $$\hat { k }$$?
Solution:
Solve like Q.No. (A)

Question 22.
Find the unit vector normal to the vectors $$\vec { a }$$ = 2$$\hat { i }$$ – $$\hat { j }$$ + $$\hat { k }$$ and $$\vec { b }$$ = 3$$\hat { i }$$ – 4$$\hat { j }$$ – $$\hat { k }$$?
Solution:
Solve like Q.No. 21 (A)

Question 23.
Find the area of parallelogram whose digonals are 3$$\hat { i }$$ + $$\hat { j }$$ – 2$$\hat { k }$$ and $$\hat { i }$$ – 3$$\hat { j }$$ + 4$$\hat { k }$$?
Solution:
ABCD is parallelogram whose diagonals are $$\vec { A }$$C = $$\vec { d }$$1 and $$\vec { B }$$D = $$\vec { d }$$2

Question 24.
By vector method prove that the square of the hypotenuse of a right angle triangle is equal to the sum of the square of the other two sides?
Solution:
Let OAB be a right angled triangle at O. Taking O as the origin. Let the position vector of $$\vec { a }$$ and $$\vec { b }$$ be a and b respectively then $$\vec { O }$$A = $$\vec { a }$$ and $$\vec { O }$$B = $$\vec { b }$$ and ∠BOA = 90°.
∴$$\vec { a }$$. $$\vec { b }$$ = 0

Question 25.
Find the moment of force 5$$\hat { i }$$ + $$\hat { k }$$ passing through the point 9$$\hat { i }$$ – $$\hat { j }$$ + 2$$\hat { k }$$ about the point 3$$\hat { i }$$ + 2$$\hat { j }$$ + $$\hat { k }$$?
Solution:

Moment of the force $$\vec { F }$$ about the point O = $$\vec { r }$$ × $$\vec { F }$$

Question 26.
(A) Prove that:

Solution:

(B) Prove that:

Question 27.
Prove that:

Question 28.
Two forces are represented by vectors $$\vec { p }$$ = 4$$\hat { i }$$ + $$\hat { j }$$ – 3$$\hat { k }$$ and $$\vec { Q }$$ = 3$$\hat { i }$$ + $$\hat { j }$$ – $$\hat { k }$$ displace a particle from points (1,2,3) to point2? (5,4,1)? Find the work done by the forces?
Solution:

Question 29.
Two forces 4$$\hat { i }$$ + 3$$\hat { j }$$ and 3$$\hat { i }$$ + 2$$\hat { j }$$ are acting on a particle, Due to the forces the particle is displaced from the point $$\hat { i }$$ + 2$$\hat { j }$$ to the point 5$$\hat { i }$$ + 4$$\hat { j }$$? Find the work done by the forces?
Solution:

Question 30.
Alorce of 6 units along the direction of vector 2$$\hat { i }$$ – 2$$\hat { j }$$ + $$\hat { k }$$ acts on a partical? The partical is displaced from point $$\hat { i }$$ + 2$$\hat { j }$$ + 3$$\hat { k }$$ to 5$$\hat { i }$$ + 3$$\hat { j }$$ + 7$$\hat { k }$$. Find the work done by the force?
Solution:
Unit vector parllel to vector 2$$\hat { i }$$ – 2$$\hat { j }$$ + $$\hat { k }$$

Question 31.
Prove that |$$\vec { a }$$ – $$\vec { b }$$ $$\vec { b }$$ – $$\vec { c }$$ $$\vec { c }$$ – $$\vec { a }$$| = 0?
Solution:
We know