First slide

Straight lines in 3D

Question

The line of intersection of the planes $x - 2 y + z - 3 = 0 a n d x + y - 2 z + 3 = 0$ is parallel to the line

Moderate

A

$x = y = z$
B

$\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$
C

$\frac{x}{1} = \frac{y}{- 2} = \frac{z}{1}$
D

$\frac{x}{1} = \frac{y}{1} = \frac{z}{- 2}$

Solution

The direction ratios of the line of intersection of two planes are proportional to the vector which is cross product of normal vectors to the planes given.
hence the direction ratios of the line of intersection of two planes is $|\begin{matrix} i & j & k \\ 1 & - 2 & 1 \\ 1 & 1 & - 2 \end{matrix}| = i (3) - j (- 3) + k (3)$
Therefore, the direction ratios of the line of intersection of two planes are proportional to the vector $i + j + k$
Hence, the equation of the line of intersection of two planes is parallel to $x = y = z$

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