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

Vector Algebra Important Questions

Vector Algebra Objective Type Questions

Question 1.
Choose the correct answer:

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\(\vec { b } \) – \(\vec { a } \)

MP Board Solutions

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\(\hat { k } \).

Answer:

  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.

MP Board Solutions

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.

Answer:

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

MP Board Solutions

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

Answer:

  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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 1
Answer:

  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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 4

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 3
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 3a

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 2
The vector having magnitude be equal to 7:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 5

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 6
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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 7

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 8

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 9

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.

MP Board Solutions

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

(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)

MP Board Solutions

(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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 13
If θ be the angle between them
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 14

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 15

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 16

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.

MP Board Solutions

Question 15.
If |\(\vec { a } \) + \(\vec { b } \)| = |\(\vec { a } \) – \(\vec { b } \)|, then prove that \(\vec { a } \) and \(\vec { b } \) are perpendicular?
Solution:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 17
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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 18

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 \(\vec { b } \)
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 19

Question 19.
Find cosine angle between vectors 2\(\hat { i } \) – 3\(\hat { j } \) + \(\hat { k } \) and \(\hat { i } \) + \(\hat { j } \) – 2\(\hat { k } \)?
Solution:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 20

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 21
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 21a
= 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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 22

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 23

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.

MP Board Solutions

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 25

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 26

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 26

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 27a
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 28

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 29

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 30

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
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 31
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.

MP Board Solutions

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
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 32
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 32a

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 = \(\frac { \vec { a } +\vec { b } +\vec { c } }{ 3 } \)
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 33

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.
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 34

MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 34a
Now position vector of a point dividing the median AD in the ratio 2 : 1 is
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 35
Position vector of a point which divides median BE in the ratio of 2 : 1 is
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 35
Position vector of a point which divides median BE in the ratio of 2 : 1 is
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 36
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
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 37

MP Board Solutions

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
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 38
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 38a
Direction cosine of vector \(\frac { 1 }{ \sqrt { 2 } } \), 0 , \(\frac { 1 }{ \sqrt { 2 } } \)
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 39

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 41

MP Board Solutions

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 42

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 43

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.

MP Board Solutions

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
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 44

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\(\hat { k } \)
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 45

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)
Answer:
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)
Answer:
λ = – \(\frac{18}{5}\)

MP Board Solutions

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 47

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 \(\frac { \theta }{ 2 } \) = \(\frac{1}{2}\) |\(\bar { a } \) – \(\bar { b } \)|
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 48

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 49
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 49a

Question 18.
In ∆ABC prove by vector method c = acosB + bcosA?
Solution:
In ∆ABC
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 50
⇒ 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
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 51
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 51a

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 52

(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)

MP Board Solutions

(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)
Answer:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 53

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)
Answer:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 54

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
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 55

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
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 56

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 57
Moment of the force \(\vec { F } \) about the point O = \(\vec { r } \) × \(\vec { F } \)
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 57a

Question 26.
(A) Prove that:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 59
Solution:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 60
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 60a

(B) Prove that:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 61

Question 27.
Prove that:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 62
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 62a

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 63

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:
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 64

MP Board Solutions

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\(\hat { j } \) + \(\hat { k } \)
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 65

Question 31.
Prove that |\(\vec { a } \) – \(\vec { b } \) \(\vec { b } \) – \(\vec { c } \) \(\vec { c } \) – \(\vec { a } \)| = 0?
Solution:
We know
MP Board Class 12th Maths Important Questions Chapter 10 Vector Algebra img 66

MP Board Class 12 Maths Important Questions