First slide

Planes in 3D

Question

$Let P (3, 2, 6) be a point in space and Q be a point on line \vec{r} = (\hat{i} - \hat{j} + 2 \hat{k}) + μ (- 3 \hat{i} + \hat{j} + 5 \hat{k}) . Then the value of μ$
$for which the vector \vec{P Q} is parallel to the plane x - 4 y + 3 z = 1 is$

Moderate

A

$\frac{1}{4}$
B

$- \frac{1}{4}$
C

$\frac{1}{8}$
D

$- \frac{1}{8}$

Solution

Any point on the line can be taken as
$\begin{aligned} Q = {(1 - 3 μ), (μ - 1), (5 μ + 2)} \\ \vec{P Q} = {- 3 μ - 2, μ - 3, 5 μ - 4) \end{aligned}$
$Now, 1 (- 3 μ - 2) - 4 (μ - 3) + 3 (5 μ - 4) = 0$
$\begin{array}{l} or - 3 μ - 2 - 4 μ + 12 + 15 μ - 12 = 0 \\ or 8 μ = 2 or μ = 1 / 4 \end{array}$

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