Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 6 Applications of Vector Algebra Ex 6.10 Textbook Questions and Answers, Notes.

## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10

Choose the most suitable answer from the given four alternatives

Question 1.
If $$\overline { a }$$ and $$\overline { b }$$ are parallel vectors, then [$$\overline { a }$$, $$\overline { c }$$, $$\overline { b }$$] is equal to
(a) 2
(b) -1
(c) 1
(d) 0
Solution:
(d) 0
Hint:
Since $$\overline { a }$$ and $$\overline { b }$$ are parallel ⇒ $$\overline { a }$$ = λ$$\overline { b }$$
[ $$\overline { a }, \overline { c }, \overline { b }$$] = [λ$$\overline { b }, \overline { c }, \overline { b }$$ ]
= λ[ $$\overline { b }, \overline { c }, \overline { b }$$ ]
= λ(0) = 0

Question 2.
If a vector $$\overline { α }$$ lies in the plane of $$\overline { ß }$$ and $$\overline { γ }$$, then
(a) [ $$\overline { α }, \overline { ß }, \overline { γ }$$ ] = 1
(b) [ $$\overline { α }, \overline { ß }, \overline { γ }$$ ] = -1
(c) [ $$\overline { α }, \overline { ß }, \overline { γ }$$ ] = 0
(d) [ $$\overline { α }, \overline { ß }, \overline { γ }$$ ] = 2
Solution:
(c) [ $$\overline { α }, \overline { ß }, \overline { γ }$$ ] = 0
Hint:
If $$\overline { α }$$ lies in $$\overline { ß }$$ & $$\overline { γ }$$ plane
we have [ $$\overline { α }, \overline { ß }, \overline { γ }$$ ] = 0

Question 3.
If $$\overline { a }$$.$$\overline { b }$$ = $$\overline { b }$$.$$\overline { c }$$ = $$\overline { c }$$.$$\overline { a }$$ = 0, then the value of [ $$\overline { a }, \overline { b }, \overline { c }$$ ] is
(a) |$$\overline { a }$$| |$$\overline { b }$$| |$$\overline { c }$$|
(b) $$\frac { 1 }{ 3 }$$|$$\overline { a }$$| |$$\overline { b }$$| |$$\overline { c }$$|
(c) 1
(d) -1
Solution:
(a) |$$\overline { a }$$| |$$\overline { b }$$| |$$\overline { c }$$|
Hint:

Question 4.
If $$\overline { a }$$, $$\overline { b }$$, $$\overline { c }$$ are three unit vectors such that $$\overline { a }$$ is perpendicular to $$\overline { b }$$ and is parallel to $$\overline { c }$$ then $$\overline { a }$$ × ($$\overline { b }$$ × $$\overline { c }$$) is equal to
(a) $$\overline { a }$$
(b) $$\overline { b }$$
(c) $$\overline { c }$$
(d) $$\overline { 0 }$$
Solution:
(b) $$\overline { b }$$
Hint:

Question 5.
If [ $$\overline { a }, \overline { b }, \overline { c }$$ ] = 1 then the value of

(a) 1
(b) -1
(c) 2
(d) 3
Solution:
(a) 1
Hint:

Question 6.
The volume of the parallelepiped with its edges represented by the vectors $$\hat { i }$$ + $$\hat { j }$$, $$\hat { i }$$ + 2$$\hat { j }$$, $$\hat { i }$$ + $$\hat { j }$$ + π$$\hat { k }$$ is
(a) $$\frac { π }{ 2 }$$
(b) $$\frac { π }{ 3 }$$
(c) π
(d) $$\frac { π }{ 4 }$$
Solution:
(c) π
Hint:
$$\left|\begin{array}{lll} 1 & 1 & 0 \\ 1 & 2 & 0 \\ 1 & 1 & \pi \end{array}\right|$$ = π$$\left|\begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right|$$
= π (2 – 1) = π

Question 7.
If $$\overline { a }$$ and $$\overline { b }$$ are unit vectors such that [$$\overline { a }$$, $$\overline { b }$$, $$\overline { a }$$ × $$\overline { b }$$] = $$\frac { 1 }{ 4 }$$, then the angle between $$\overline { a }$$ and $$\overline { b }$$ is
(a) $$\frac { π }{ 6 }$$
(b) $$\frac { π }{ 4 }$$
(c) $$\frac { π }{ 3 }$$
(d) $$\frac { π }{ 2 }$$
Solution:
(a) $$\frac { π }{ 6 }$$
Hint:

Question 8.
If $$\overline { a }$$ = $$\hat { i }$$ + $$\hat { j }$$ + $$\hat { k }$$, $$\overline { b }$$ = $$\hat { i }$$ + $$\hat { j }$$, $$\overline { c }$$ = $$\hat { i }$$ and ($$\overline { a }$$ × $$\overline { b }$$)$$\overline { c }$$ – λ$$\overline { a }$$ + µ$$\overline { b }$$ then the value of λ + µ is
(a) 0
(b) 1
(c) 6
(d) 3
Solution:
(a) 0
Hint:
$$\overline { a }$$.$$\overline { c }$$ = 1 and $$\overline { b }$$.$$\overline { c }$$ = 1
($$\overline { a }$$ × $$\overline { b }$$)$$\overline { c }$$ = ($$\overline { c }$$ × $$\overline { a }$$)$$\overline { b }$$ – ($$\overline { c }$$ × $$\overline { b }$$)$$\overline { a }$$ = λ$$\overline { a }$$ + µ$$\overline { b }$$
⇒ µ = c; a = 1λ = -($$\overline { c }$$.$$\overline { b }$$) = -1
µ + λ = 1 – 1 = 0

Question 9.
If $$\overline { a }$$, $$\overline { b }$$, $$\overline { c }$$ are non-coplanar, non-zero vectors
such that [$$\overline { a }$$, $$\overline { b }$$, $$\overline { c }$$] = 3, then {[$$\overline { a }$$ × $$\overline { b }$$, $$\overline { b }$$ × $$\overline { c }$$, $$\overline { c }$$ × $$\overline { a }$$]²} is equal to
is equal to
(a) 81
(b) 9
(c) 27
(d) 18
Solution:
(a) 81
Hint:

= 34 = 81

Question 10.
If $$\overline { a }$$, $$\overline { b }$$, $$\overline { c }$$ are three non-coplanar vectors such that $$\overline { a }$$ × ($$\overline { b }$$ × $$\overline { c }$$) = $$\frac { \overline{b}+\overline{c} }{ √2 }$$ then the angle between $$\overline { a }$$ and $$\overline { b }$$ is
(a) $$\frac { π }{ 2 }$$
(b) $$\frac { 3π }{ 4 }$$
(c) $$\frac { π }{ 4 }$$
(d) π
Solution:
(b) $$\frac { 3π }{ 4 }$$
Hint:

Question 11.
If the volume of the parallelepiped with $$\overline { a }$$ × $$\overline { b }$$, $$\overline { b }$$ × $$\overline { c }$$, $$\overline { c }$$ × $$\overline { a }$$ as coterminous edges is 8 cubic units, then the volume of the parallelepiped with ($$\overline { a }$$ × $$\overline { b }$$) × ($$\overline { b }$$ × $$\overline { c }$$), ($$\overline { b }$$ × $$\overline { c }$$) × ($$\overline { c }$$ × $$\overline { a }$$) and ($$\overline { c }$$ × $$\overline { a }$$) × ($$\overline { a }$$ × $$\overline { b }$$) as coterminous edges is
(a) 8 cubic units
(b) 512 cubic units
(c) 64 cubic units
(d) 24 cubic units
Solution:
(c) 64 cubic units
Hint:
Given volume of the parallelepiped with

Question 12.
Consider the vectors $$\overline { a }$$, $$\overline { b }$$, $$\overline { c }$$, $$\overline { d }$$ such that ($$\overline { a }$$ × $$\overline { b }$$) × ($$\overline { c }$$ × $$\overline { d }$$) = $$\overline { 0 }$$ Let P1 and P2 be the planes determined by the pairs of vectors $$\overline { a }$$, $$\overline { b }$$ and $$\overline { c }$$, $$\overline { d }$$ respectively. Then the angle between P1 and P2 is
(a) 0°
(b) 45°
(c) 60°
(d) 90°
Solution:
(a) 0°
Hint:
A vector perpendicular to the plane P1 of a, b is $$\overline { a }$$ × $$\overline { b }$$,
A vector perpendicular to the plane P2 of c and d is $$\overline { c }$$ × $$\overline { d }$$
∴ ($$\overline { a }$$ × $$\overline { b }$$) × ($$\overline { c }$$ × $$\overline { d }$$) = 0
⇒ ($$\overline { a }$$ × $$\overline { b }$$) || $$\overline { c }$$ × $$\overline { d }$$
⇒ The angle between the planes is $$\overline { 0 }$$

Question 13.
If $$\overline { a }$$ × ($$\overline { b }$$ × $$\overline { c }$$) = ($$\overline { a }$$ × $$\overline { b }$$) × $$\overline { c }$$ where $$\overline { a }$$, $$\overline { b }$$, $$\overline { c }$$ are any three vectors such that $$\overline { b }$$.$$\overline { c }$$ ≠ 0 and $$\overline { a }$$.$$\overline { b }$$ ≠ 0, then $$\overline { a }$$ and $$\overline { c }$$ are
(a) perpendicular
(b) parallel
(c) inclined at angle $$\frac { π }{ 3 }$$
(d) inclined at an angle $$\frac { π }{ 6 }$$
Solution:
(b) parallel
Hint:

Question 14.
If $$\overline { a }$$ = 2$$\hat { i }$$ + 3$$\hat { j }$$ – $$\hat { k }$$, $$\overline { b }$$ = $$\hat { i }$$ + 2$$\hat { j }$$ – 5$$\hat { k }$$, $$\overline { c }$$ = 3$$\hat { i }$$ + 5$$\hat { j }$$ – $$\hat { k }$$ then $$\overline { a }$$ vector perpendicular to a and lies in the plane containing $$\overline { b }$$ and $$\overline { c }$$ is
(a) -17$$\hat { i }$$ + 21$$\hat { j }$$ – 97$$\hat { k }$$
(b) 17$$\hat { i }$$ + 21$$\hat { j }$$ – 123$$\hat { k }$$
(c) -17$$\hat { i }$$ – 21$$\hat { j }$$ + 97$$\hat { k }$$
(d) -17$$\hat { i }$$ – 21$$\hat { j }$$ – 97$$\hat { k }$$
Solution:
(d) -17$$\hat { i }$$ – 21$$\hat { j }$$ – 97$$\hat { k }$$
Hint:
A vector ⊥r to $$\overline { a }$$ and lies in the plane containing $$\overline { b }$$ and $$\overline { c }$$

Question 15.
The angle between the lines $$\frac { x-2 }{ 3 }$$ = $$\frac { y+1 }{ -2 }$$, z = 2 and $$\frac { x-1 }{ 1 }$$ = $$\frac { 2y+3 }{ 3 }$$ = $$\frac { z+5 }{ 2 }$$ is
(a) $$\frac { π }{ 6 }$$
(b) $$\frac { π }{ 4 }$$
(c) $$\frac { π }{ 3 }$$
(d) $$\frac { π }{ 2 }$$
Solution:
(d) $$\frac { π }{ 2 }$$
Hint:

Question 16.
If the line $$\frac { x-2 }{ 3 }$$ = $$\frac { y-1 }{ -5 }$$ = $$\frac { z+2 }{ 2 }$$ lies in the plane x + 3y – αz + ß = 0 then (α + ß) is
(a) (-5, 5)
(b) (-6, 7)
(c) (5, -5)
(d) (6, -7)
Solution:
(b) (-6, 7)
Hint:
$$\frac { x-2 }{ 3 }$$ = $$\frac { y-1 }{ 5 }$$ = $$\frac { z+2 }{ 2 }$$ = λ ⇒ (3λ + 2, -5λ + 1, 2λ – 2)
which lie in x + 3y – αz + ß = 0
(3λ + 2) + 3(-5λ + 1) – α(2λ – 2) + ß = 0
3λ + 2 – 15λ + 3 – 2αλ + 2α + ß = 0.
(-12λ – 2αλ) + 2α + ß + 5 = 0.
-12λ – 2αλ = 0
2αλ = -12λ
α = -6
2α+ ß +5 = 0
-12 + ß + 5 = 0
ß – 7 = 0
ß = 7
(α, ß) = (-6, 7)

Question 17.
The angle between the line $$\overline { r }$$ = ($$\hat { i }$$ + 2$$\hat { j }$$ – 3$$\hat { k }$$) + t(2$$\hat { i }$$ + $$\hat { j }$$ – 2$$\hat { k }$$) and the plane $$\overline { r }$$ ($$\hat { i }$$ + $$\hat { j }$$) + 4 = 0 is
(a) 0°
(b) 30°
(c) 45°
(d) 90°
Solution:
(c) 45°
Hint:

Question 18.
The co-ordinates of the point where the line $$\overline { r }$$ = (6($$\hat { i }$$ – $$\hat { j }$$ – 3$$\hat { k }$$) + t(-$$\hat { i }$$ + $$\hat { k }$$ meets the plane $$\overline { r }$$ (($$\hat { i }$$ + ($$\hat { j }$$ – ($$\hat { k }$$) = 3 are
(a) (2, 1, 0)
(b) (7, -1, -7)
(c) (1, 2, -6)
(d) (5, -1, 1)
Solution:
(d) (5, -1, 1)
Hint:
Given $$\overline { r }$$ = (6($$\hat { i }$$ – ($$\hat { j }$$ – 3($$\hat { k }$$) + t(-($$\hat { i }$$ + ($$\hat { k }$$)
$$\frac { x-6 }{ -1 }$$ = $$\frac { y+1 }{ 0 }$$ = $$\frac { z+3 }{ 4 }$$ = t ⇒ (-t + 6, -1, 4t – 3)
which meets x + y – z = 3
-t + 6 – 1 – 4t + 3 = 3
-5t + 5 = 0
5t = 5
t = 1
∴ Co-ordinate is (5, -1, 1)

Question 19.
Distance from the origin to the plane 3x – 6y + 2z + 7 = 0 is
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(b) 1
Hint:
(x1, y1, z1) = (o, 0, o)
(a, b, c) = (3, -6, 2); d = 7.
d = $$\frac { ax_1+by_1+cz_1+d }{ \sqrt{a^2+b^2+c^2} }$$ = $$\frac { 7 }{ \sqrt{9+36+4} }$$ = $$\frac { 7 }{ 7 }$$ = 1

Question 20.
The distance between the planes
x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0 is
(a) $$\frac { √7 }{ 2√2 }$$
(b) $$\frac { 7 }{ 2 }$$
(c) $$\frac { √7 }{ 2 }$$
(d) $$\frac { 7 }{ 2√2 }$$
Solution:
(a) $$\frac { √7 }{ 2√2 }$$
Hint:
x + 2y + 3z+7 = 0
2x + 4y + 6z + 7 = 0
(÷ 2) x + 2y + 3z + $$\frac { 7 }{ 2 }$$ = 0
(1) and (2) are parallel planes

Question 21.
If the direction cosines of a line are $$\frac { 1 }{ c }$$, $$\frac { 1 }{ c }$$, $$\frac { 1 }{ c }$$
(a) c = ±3
(b) c = ±√3
(c) c > 0
(d) 0 < c < 1
Solution:
(b) c = ±√3
Hint:
cos²α + cos²ß + cos²γ = 1
$$\frac { 1 }{ c^2 }$$ + $$\frac { 1 }{ c^2 }$$ + $$\frac { 1 }{ c^2 }$$ = 1
$$\frac { 3 }{ c ^2}$$ = 1
c² = 3
c = ±√3

Question 22.
The vector equation $$\overline { r }$$ = ($$\hat { i }$$ – 2$$\hat { j }$$ – $$\hat { k }$$) + t(6$$\hat { i }$$ – $$\hat { k }$$) represents a straight line passing through the points
(a) (0, 6, -1) and (1, -2, -1)
(b) (0, 6, -1) and (-1, -4, -2)
(c) (1, -2, -1) and (1, 4, -2)
(d) (1, -2, -1) and (0, -6, 1)
Solution:
(c) (1, -2, -1) and (1, 4, -2)
Hint:
Given vector equation is

Question 23.
If the distance of the point (1, 1, 1) from the origin is half of its distance from the plane x + y + z + k = Q, then the values of k are
(a) ±3
(b) ±6
(c) -3, 9
(d) 3, -9
Solution:
(d) 3, -9
Hint:

Question 24.
If the planes $$\overline { r }$$ (2$$\hat { i }$$ – λ$$\hat { j }$$ + $$\hat { k }$$) = 3 and $$\overline { r }$$ (4$$\hat { i }$$ + $$\hat { j }$$ – µ$$\hat { k }$$) = 5 are parallel, then the value of λ and µ are
(a) $$\frac { 1 }{ 2 }$$, -2
(b) –$$\frac { 1 }{ 2 }$$, 2
(c) –$$\frac { 1 }{ 2 }$$, -2
(d) $$\frac { 1 }{ 2 }$$, 2
Solution:
(c) –$$\frac { 1 }{ 2 }$$, -2
Hint:

Question 25.
If the length of the perpendicular from the origin to the plane 2x + 3y + λz = 1, λ > 0 is $$\frac { 1 }{ 5 }$$, then the value of λ is
(a) 2√3
(b) 3√2
(c) 0
(d) 1
Solution:
(a) 2√3
Hint:

5 = $$\sqrt { 4+9+λ^2 }$$
25 = 4 + 9 + λ²
25 = 13 + λ²
λ² = 12
λ = 2√3