First slide

Planes in 3D

Question

The angle between $\hat{i}$ line of the intersection of the plane $\vec{r} \cdot (\hat{i} + 2 \hat{j} + 3 \hat{k}) = 0 and \vec{r} \cdot (3 \hat{i} + 3 \hat{j} + \hat{k}) = 0$ is

Moderate

A

$\cos^{- 1} (\frac{1}{3})$
B

$\cos^{- 1} (\frac{1}{\sqrt{3}})$
C

$\cos^{- 1} (\frac{2}{\sqrt{3}})$
D

none of these

Solution

Line of intersection of $\vec{r} \cdot (\hat{i} + 2 \hat{j} + 3 \hat{k}) = 0 and \vec{r} \cdot (3 \hat{i} + 3 \hat{j} + \hat{k}) = 0$ will be Parallel to
$(3 \hat{i} + 3 \hat{j} + \hat{k}) \times (\hat{i} + 2 \hat{j} + 3 \hat{k}), i.e., 7 \hat{i} - 8 \hat{j} + 3 \hat{k}$
If the required angle is $θ$ then
$\cos θ = \frac{7}{\sqrt{49 + 64 + 9}} = \frac{7}{\sqrt{122}}$

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