Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2

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Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2

Question 1.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 1
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 2

Question 2.
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 3.
Solution:
Volume of the parallelepiped = \(\| \vec{a}, \vec{b}, \vec{c}]\)
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 4
= -264 + 224 + 760 = 720 cubic units

Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2

Question 3.
The volume of the parallelepiped whose coterminus edges are Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 5, \(-3 \vec{i}+7 \vec{j}+5 \vec{k}\) is 90 cubic units. Find the value of λ
Solution:
Given, Volume of the parallelepiped = 90 cubic units
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 6

Question 4.
If \(\vec{a}, \vec{b}, \vec{c}\) are three non-coplanar vectors represented by concurrent edges of a parallelepiped of volume 4 cubic units, find the value of Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 7
Solution:
Let \(\vec{a}, \vec{b}, \vec{c}\) be the concurrent edges of parallelepiped
Given volume of parallelepiped = 4 cubic units
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 8

Question 5.
Find the altitude of a parallelepiped determined by the vectors \(\vec{a}=-2 \hat{i}+5 \hat{j}+3 \hat{k}\), \(\hat{b}=\hat{i}+3 \hat{j}-2 \hat{k}\) and \(\vec{c}=-3 \vec{i}+\vec{j}+4 \vec{k}\) if the base is taken as the parallelogram determined by b and c.
Solution:
Volume = Base Area × Height
\(|[\vec{a}, \vec{b}, \vec{c}]|=|\vec{b} \times \vec{c}|\) × Height
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 9

Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2

Question 6.
Determine whether the three vectors \(2 \hat{i}+3 \hat{j}+\hat{k}, \hat{i}-2 \hat{j}+2 \hat{k}\) and \(3 \hat{i}+\hat{j}+3 \hat{k}\) are coplanar.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 10

Question 7.
Let Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 11 If c1 = 1 and c2 = 2, find c3 such that \(\vec{a}, \vec{b}\) and \(\vec{c}\) and c are coplanar.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 12

Question 8.
If Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 13, show that \([\vec{a} \vec{b} \vec{c}]\) depends neither x nor y.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 14

Question 9.
If the vectors
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 15
are coplanar, prove that c is the geometric mean of a and b.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 16
∴ c is the geometric means of ‘a’ and ‘b’.

Question 10.
Let \(\vec{a}, \vec{b}, \vec{c}\) be three non-zero vectors such that \(\vec{c}\) is a unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\). If the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{6}\) show that Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 17 .
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 18
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 19

Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 Additional Problems

Question 1.
If the edges Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 20 meet a vertex, find the volume of the parallelepiped.
Solution:
Volume of the parallelepiped = Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 21
The volume cannot be negative
∴ Volume of parallelepiped = 264 cu. units

Question 2.
If Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 22 and \(\vec{x} \neq \overrightarrow{0}\) then show that \(\vec{a}, \vec{b}, \vec{c}\) are coplanar.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 23

Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2

Question 3.
The volume of a parallelepiped whose edges are represented by \(-12 \vec{i}+\lambda k\), \(3 \vec{j}-\vec{k}, 2 \vec{i}+\vec{j}-15 \vec{k}\) is 546. Find the value of λ.
Solution:
Volume of the parallelepiped = Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 24
= -12 [-45 + 1] – 0 () + λ [0 – 6] = -12 (-44) -6 λ
= 528 – 6λ = 546 (given)
⇒ -6λ = 546 – 528 = 18
∴ λ = \(\frac{18}{-6}\) = -3

Question 4.
Prove that \(|\vec{a} \vec{b} \vec{c}|\) = abc if and only if \(\vec{a}, \vec{b}, \vec{c}\) are mutually perpendicular.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 25
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 26

Question 5.
Show that the points (1, 3, 1), (1, 1, -1), (-1, 1, 1), (2, 2, -1) are lying on the same plane.
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 27

Question 6.
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 28
Solution:
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 29
Samacheer Kalvi 12th Maths Solutions Chapter 6 Applications of Vector Algebra Ex 6.2 30

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