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

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

Choose the correct or the most suitable answer from the given four alternatives:
Question 1.
If \(\vec{a}\) and \(\vec{b}\) are parallel vector, then \([\vec{a}, \vec{c}, \vec{b}]\) is equal to ………………..
(a) 2
(b) -1
(c) 1
(d) 0
Solution:
(d) 0
Hint:
\(\vec{a}\) and \(\vec{b}\) are parallel vectors, so \(\vec{a} \times \vec{b}\) = 0
then \([\vec{a} \vec{c} \vec{b}]=-[\vec{a} \vec{b} \vec{c}]=-(\vec{a} \times \vec{b}) \cdot \vec{c}\) = 0

Question 2.
If a vector \(\vec{\alpha}\) lies in the plane of \(\vec{\beta}\) and \(\vec{\gamma}\), then …………
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 1
Solution:
(c) \([\vec{\alpha}, \vec{\beta}, \vec{\gamma}]\) = 0
Hint:
Since \([\vec{\alpha}, \vec{\beta}, \vec{\gamma}]\) are lie in the same plane
so \([\vec{\alpha}, \vec{\beta}, \vec{\gamma}]\) = 0

Question 3.
If \(\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{c}=\vec{c} \cdot \vec{a}\)= 0, then the value of \(|[\vec{a}, \vec{b}, \vec{c}]|\) is ……………….
(a) \(|\vec{a}||\vec{b}||\vec{c}|\)
(b) \(\frac{1}{3}|\vec{a}||\vec{b}||\vec{c}|\)
(c) 1
(d) -1
Solution:
(a) \(|\vec{a}||\vec{b}||\vec{c}|\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 2

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

Question 4.
If \(\vec{a}, \vec{b}, \vec{c}\) are three unit vectors such that \(\vec{a}\) is perpendicular to \(\vec{b}\), and is parallel to \(\vec{c}\) then \(\vec{a} \times(\vec{b} \times \vec{c})\)is equal to ………………….
(a) \(\vec{a}\)
(b) \(\vec{b}\)
(c) \(\vec{c}\)
(d) \(\vec{0}\)
Solution:
(b) \(\vec{b}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 3

Question 5.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 4
(a) 1
(b) -1
(c) 2
(d) 3
Solution:
(a) 1
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 5

Question 6.
The volume of the parallelepiped with its edges represented by the vectors \(\hat{i}+\hat{j}, i+2 \hat{j}\), \(\hat{i}+\hat{j}+\pi \hat{k}\) is ……………
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 6
Solution:
(c) π
Hint:
Volume = \([\vec{a} \vec{b} \vec{c}]=\left|\begin{array}{ccc}{1} & {1} & {0} \\ {1} & {2} & {0} \\ {1} & {1} & {\pi}\end{array}\right|\)
= π(2 – 1) = π cubic units

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

Question 7.
If \(\vec{a}\) and \(\vec{b}\) are unit vectors such that \([\vec{a}, \vec{b}, \vec{a} \times \vec{b}]=\frac{\pi}{4}\) then the angle between \(\vec{a}\) and \(\vec{b}\) is …………
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 7
Solution:
(a) \(\frac{\pi}{6}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 8

Question 8.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 9
(a) 0
(b) 1
(c) 6
(d) 3
Solution:
(a) 0
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 10
Equate corresponding coefficients on both sides
λ + µ = 0 and λ = -1 this gives µ = 1
∴ Then the value of λ + µ = 0.

Question 9.
If \(\vec{a}, \vec{b}, \vec{c}\) are three non-coplanar vectors such that \(\vec{a} \times(\vec{b} \times \vec{c})=\frac{\vec{b}+\vec{c}}{\sqrt{2}}\), then the angle between \(\vec{a}\) and \(\vec{b}\) is …………
(a) 81
(b) 9
(c) 27
(d) 18
Solution:
(a) 81
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 11

Question 10.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 12
Solution:
(b) \(\frac{3 \pi}{4}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 13

Question 11.
If the volume of the parallelepiped with \(\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}\) as coterminous edges is 8 cubic units, then the volume of the parallelepiped with \((\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c}),(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a})\) and \((\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b})\) 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:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 14

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

Question 12.
Consider the vectors \(\vec{a}, \vec{b}, \vec{c}, \vec{d}\) such that \((\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})=\overrightarrow{0}\). Let P1 and P2 be the planes determined by the pairs of vectors \(\vec{a}, \vec{b}\) and \(\vec{c}, \vec{d}\) respectively. Then the angle between P1 and P2 is ……………..
(a) 0°
(b) 45°
(c) 60°
(d) 90°
Solution:
(a) 0°
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 15

Question 13.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 16
(a) perpendicular
(b) parallel
(c) inclined at an angle \(\frac{\pi}{3}\)
(d) inclined at an angle \(\frac{\pi}{6}\)
Solution:
(b) parallel
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 17

Question 14.
If Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 18, then a vector perpendicular to \(\vec{a}\) and lies in the plane containing \(\vec{b}\) and \(\vec{c}\) is
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 19
Solution:
(d) \(-17 \hat{i}-21 \hat{j}-97 \hat{k}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 20

Question 15.
The angle between the lines Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 21 is ………..
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 22
Solution:
(d) \(\frac{\pi}{2}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 23
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 24

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

Question 16.
If the line Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 25 lies in the plane x + 3 + αz + β = 0, then (α, β) is …………..
(a) (-5, 5)
(b) (-6, 7)
(c) (5, -5)
(d) (6, -7)
Solution:
(d) (-6, 7)
Hint:
d.c.s of the first line = (3, -5, 2)
d.c.s of the line perpendicular to plane = (1, 3, -α)
a1a2 + b1b2 + c1c2 = 0
3 – 15 – 2α = 0 => -12 – 2α = 0
-2α =12 => α = -6
Plane passes through the point (2, 1, -2) so
2 + 3 + 6(-2) + β = 0 => β = 7
(α, β) = (-6, 7)

Question 17.
The angle between the line \(\vec{r}=(\hat{i}+2 \hat{j}-3 \hat{k})+t(2 \hat{i}+\hat{j}-2 \hat{k})\) and the plane \(\vec{r} \cdot(\hat{i}+\hat{j})+4\) = 0 is ……………
(a) 0°
(b) 30°
(c) 45°
(d) 90°
Solution:
(c) 45°
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 26

Question 18.
The coordinates of the point where the line \(\vec{r}=(6 \hat{i}-\hat{j}-3 \hat{k})+t(-\hat{i}+4 \hat{k})\) meets the plane \(\vec{r} \cdot(\hat{i}+\hat{j}-\hat{k})\) are
(a) (2, 1, 0)
(b) (7, -1, -7)
(c) (1, 2, -6)
(c) (5, -1, 1)
Solution:
(d) (5, -1, 1)
Hint:
Cartesian equation of the line
\(\frac{x-6}{-1}=\frac{y+1}{0}+\frac{z+3}{4}\) = λ
(-λ + 6, -1, 4λ – 3)
This meets the plane x + y – z = 3
-λ + 6 – 1 – 41 + 3 = 3 ⇒ -5λ = -5
λ = 1
The required point (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:
Distance from the origin (0, 0, 0) to the plane
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 27

Question 20.
The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0 is ……………
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 28
Solution:
(a) \(\frac{\sqrt{7}}{2 \sqrt{2}}\)
Hint:
x + 2y + 3z + 1 = 0; 2x + 4y + 6z + 7 = 0
Multiplying 2 on both sides
2x + 4y + 6z + 14 = 0 .
a = 2, b = 4, c = 6, d1 = 14, d2 = ?
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 29

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

Question 21.
If the direction cosines of a line are \(\frac{1}{c}, \frac{1}{c}, \frac{1}{c}\), then ……………….
(a) c = ±3
(b) c = ± \(\sqrt{3}\)
(c) c > 0
(d) 0 < c < 1
Solution:
(b) c = ± \(\sqrt{3}\)
Hint:
We know that sum of the squares of direction cosines = 1
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 30

Question 22.
The vector equation \(\vec{r}=(\hat{i}-2 \hat{j}-\hat{k})+t(6 \vec{j}-\hat{k})\) 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:
The required vector equation is \(\vec{r}=\vec{a}+t(\vec{b}-\vec{a})\)
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 31
From (1) and (2) The points are (1, -2, -1) and (1, 4, -2)

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= 0, then the values of k are ………….. (a) ±3 (b) ±6 (c) -3, 9 (d) 3, -9
Solution:
(d) 3, -9
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 32

Question 24.
If the planes \(\vec{r} \cdot(2 \hat{i}-\lambda \hat{j}+\hat{k})=3 \text { and } \vec{r} \cdot(4 \hat{i}+\hat{j}-\mu \hat{k})\) = 5 are parallel, then the value of λ and µ are ……………
Solution:
(c) \(-\frac{1}{2}\), -2
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 33

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) \(2 \sqrt{3}\)
(b) \(3 \sqrt{2}\)
(c) 0
(d) 1
Solution:
(a) \(2 \sqrt{3}\)
Hint:
Given length of perpendicular from origin to the plane = \(-\frac{1}{5}\)
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 34

Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 Additional Problems

Question 1.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 35
(a) 6
(b) 10
(c) 12
(d) 24
Solution:
(c) 12
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 36

Question 2.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 37
(a) only x
(b) only y
(c) Neither x nor y
(d) Both x and y
Solution:
(c) Neither x nor y
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 38

Question 3.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 39
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(b) 3
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 40

Question 4.
The value of Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 41 = ………………
(a) 1
(b) 3
(c) -3
(d) 0
Solution:
(b) 3
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 42

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

Question 5.
Let a, b, c be distinct non-negative numbers. If the vectors Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 43 lie in a plane, then c is ……………..
(a) the A.M. of a and b
(b) the G.M. of a and b
(c) the H.M. of a and b
(d) equal to zero.
Solution:
(b) the G.M. of a and b
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 44

Question 6.
The value of \(\hat{i} \cdot(\hat{j} \times \hat{k})+(\hat{i} \times \hat{k}) \cdot \hat{j}\) ………….
(a) 1
(b) -1
(c) 0
(d) \(\hat{j}\)
Solution:
(c) 0
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 45

Question 7.
The value of \((\hat{i}-\hat{j}, \hat{j}-\hat{k}, \hat{k}-\hat{i})\) is …………..
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(a) 0
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 46
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 47

Question 8.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 48
Solution:
(c) \(\vec{u}=\overrightarrow{0}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 49

Question 9.
The area of the parallelogram having a diagonal \(3 \vec{i}+\vec{j}-\vec{k}\) and a side \(\vec{i}-3 \vec{j}+4 \vec{k}\) is ………………
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 50
Solution:
(d) \(3 \sqrt{30}\)
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 51

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

Question 10.
If Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 52, then ……………….
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 53
Solution:
(d) \(\vec{x}=\overrightarrow{0} \text { or } \vec{y}=\overrightarrow{0} \text { or } \vec{x} \text { and } \vec{y}\) are parallel
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 54

Question 11.
If \(\overrightarrow{\mathrm{PR}}=2 \vec{i}+\vec{j}+\vec{k}, \overrightarrow{\mathrm{Question}}=-\vec{i}+3 \vec{j}+2 \vec{k}\), then the area of the quadrilateral PQRS is ………………..
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 55
Solution:
(c) \(\frac{5 \sqrt{3}}{2}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 56

Question 12.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 57
Solution:
(c) \(\vec{c}\) parallel to \(\vec{a}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.10 58

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