The angle between planes 2x - y + z = 6 and x + y + 2z = 3 is

Correct Answer is : 60∘60^{}60∘

Correct Answer is : 60∘60^{}60∘

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CGPET 2014 Solved Paper

Section: Mathematics

Question : 101 of 150

Marks: +1, -0

2x - y + z = 6

x + y + 2z = 3

Solution:

Given equations of plane are

2x - y + z = 6 \;\text{ and }\; x + y + 2z = 3

On comparing with

ax + by + cz = d

, we get

\;\text{ and }\;\;\;\;a_1=2,\;b_1=-1,\;c_1=1

\; a_2=1,\; b_1=1,\; c_1=2

\therefore \;\;

Angle between two planes is

\theta = \cos^{-1} \left( \; \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} \right)

= \cos^{-1} \left( \; \frac{2 \times 1 + (-1) \times 1 + 1 \times 2}{\sqrt{2^2 + (-1)^2 + 1^2} \sqrt{1^2 + 1^2 + 2^2}} \right)

= \cos^{-1} \left( \; \frac{2-1+2}{\sqrt{4+1+1} \sqrt{1+1+4}} \right)

= \cos^{-1} \left( \; \frac{3}{\sqrt{6} \times \sqrt{6}} \right) = \cos^{-1} \left( \; \frac{3}{6} \right) = \cos^{-1} \left( \; \frac{1}{2} \right)

= 60^{\circ}

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