First slide

Planes in 3D

Question

The equation of the plane passing through a point with position vector $\hat{i} + 2 \hat{j} + 3 \hat{k}$ and parallel to the plane $\vec{r} . (3 \hat{i} + 4 \hat{j} + 5 \hat{k}) = 0$ is

Moderate

A

$3 x + 4 y - 5 z + 26 = 0$
B

$3 x + 4 y + 5 z - 26 = 0$
C

$3 x - 4 y + 5 z - 26 = 0$
D

$3 x + 4 y - 5 z - 26 = 0$

Solution

Equation of the plane parallel to $\bar{r} . (3 \bar{i} + 4 \bar{j} + 5 \bar{k}) = 0$ is given by $(x \bar{i} + y \bar{j} + z \bar{k}) (3 \bar{i} + 4 \bar{j} + 5 \bar{k}) + λ = 0$
It passes through (1,2,3)
$∴ λ = - 26$

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