First slide

Planes in 3D

Question

The image of the point having position vector $\bar{i} + 3 \bar{j} + 4 \bar{k}$ in the plane $\bar{r} . (2 \bar{i} - \bar{j} + 4 \bar{k}) + 3 = 0$

Moderate

A

$- 3 \bar{i} + 5 \bar{j} + \bar{k}$
B

$- 3 \bar{i} + 5 \bar{j} + 2 \bar{k}$
C

$3 \bar{i} + 5 \bar{j} + \bar{k}$
D

$3 \bar{i} + 5 \bar{j} + 2 \bar{k}$

Solution

$(x_{1}, y_{1}, z_{1}) = (1, 3, 4)$ plane $2 x - y + z + 3 = 0$
If $(h, k, l)$ in the image of $(x_{1}, y_{1}, z_{1})$
$\Rightarrow \frac{h - x_{1}}{a} = \frac{k - y_{1}}{b} = \frac{l - z_{1}}{c}$
$= \frac{- 2 (a x_{1} + b y_{1} + c z_{1} + d)}{a^{2} + b^{2} + c^{2}}$

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