13 In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

      (b)      2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

Answers (1)

Two planes

L_{1}:a_{1}x+b_{1}y+c_{1}z = 0 whose direction ratios are a_{1},b_{1},c_{1} and L_{2}:a_{2}x+b_{2}y+c_{2}z = 0 whose direction ratios are a_{2},b_{2},c_{2},

are said to Parallel: 

If, \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}

and Perpendicular:

If, a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2} = 0

And the angle between L_{1}\ and\ L_{2} is given by the relation,

A = \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 |
So, given two planes  2x + y + 3z -2 = 0\ and\ x -2y + 5 = 0

Here,

a_{1} = 2,b_{1} = 1, c_{1} = 3   and   a_{2} = 1,b_{2} = -2, c_{2} = 0

So, applying each condition to check:

Perpendicular check: a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2} = 0

\Rightarrow 2(1)+1(-2)+3(0) = 2-2+0 = 0.

Thus, the given planes are perpendicular to each other.

 

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