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.

Solution:

Question 2.
Find the volume of the parallelepiped whose coterminous edges are represented by the vectors .
Solution:
Volume of the parallelepiped = \(\| \vec{a}, \vec{b}, \vec{c}]\)

= -264 + 224 + 760 = 720 cubic units

Question 3.
The volume of the parallelepiped whose coterminus edges are , \(-3 \vec{i}+7 \vec{j}+5 \vec{k}\) is 90 cubic units. Find the value of λ
Solution:
Given, Volume of the parallelepiped = 90 cubic units

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
Solution:
Let \(\vec{a}, \vec{b}, \vec{c}\) be the concurrent edges of parallelepiped
Given volume of parallelepiped = 4 cubic units

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

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:

Question 7.
Let If c₁ = 1 and c₂ = 2, find c₃ such that \(\vec{a}, \vec{b}\) and \(\vec{c}\) and c are coplanar.
Solution:

Question 8.
If , show that \([\vec{a} \vec{b} \vec{c}]\) depends neither x nor y.
Solution:

Question 9.
If the vectors

are coplanar, prove that c is the geometric mean of a and b.
Solution:

∴ 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 .
Solution:

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

Question 1.
If the edges meet a vertex, find the volume of the parallelepiped.
Solution:
Volume of the parallelepiped =
The volume cannot be negative
∴ Volume of parallelepiped = 264 cu. units

Question 2.
If and \(\vec{x} \neq \overrightarrow{0}\) then show that \(\vec{a}, \vec{b}, \vec{c}\) are coplanar.
Solution:

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 =
= -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:

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:

Question 6.

Solution: