First slide

Direction ratios and direction cosines

Question

Let $L$ be the line of intersection of the planes $2 x + 3 y + z = 1$ and $x + 3 y + 2 z = 2$ . If $L$ makes an angle $α$ with the positive $x -$ axis then $\cos α$ is

Easy

A

$\frac{1}{\sqrt{3}}$
B

$\frac{1}{2}$
C

$1$
D

$\frac{1}{\sqrt{2}}$

Solution

The line of intersection of two planes is perpendicular to the both normal vectors.
The normal vectors to the given planes are $\bar{n_{1}} = 2 i + 3 j + k,$ and $\bar{n_{2}} = i + 3 j + 2 k,$
The vector along line of intersection of two planes is $\bar{n_{1}} \times \bar{n_{2}} = |\begin{matrix} i & j & k \\ 2 & 3 & 1 \\ 1 & 3 & 2 \end{matrix}| = i (3) - j (3) + k (3)$
Hence the direction cosines of line are of intersection of two plane is $〈\frac{3}{\sqrt{9 + 9 + 9}}, - \frac{3}{\sqrt{9 + 9 + 9}}, \frac{3}{\sqrt{9 + 9 + 9}}〉 = 〈\frac{1}{\sqrt{3}}, - \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}〉$
Therefore, $\cos α = \frac{1}{\sqrt{3}}$

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