First slide

Straight lines in 3D

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 positive $x -$ axis, then $\cos α$ equals.

Moderate

A

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

$\frac{1}{2}$
C

$- 1$
D

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

Solution

The given planes are $2 x + 3 y + z = 1$ and $x + 3 y + 2 z = 2$
The line of intersection of those two planes is along the vector $\bar{n} = |\begin{matrix} i & j & k \\ 2 & 3 & 1 \\ 1 & 3 & 2 \end{matrix}| = i (3) - j (3) + k (3)$
Hence, the direction ratios of the line of intersection of two planes is $⟨ 1, - 1, 1 ⟩$
The angle made by the line with $x -$ axis is $α$ then $\cos α = \pm \frac{1}{\sqrt{3}}$

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