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Angle between Two Planes

Let A0,0,0,...

Question

Let $A (0, 0, 0), B (1, 1, 1), C (3, 2, 1)$ and $D (3, 1, 2)$ be four points. The angle between the planes through the points $A, B, C$ and through the points $A, B, D$ is

\frac{π}{2}

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\frac{π}{6}

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\frac{π}{4}

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\frac{π}{3}

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Solution

The correct option is A $\frac{π}{3}$
Let $n_{1}$ and $n_{2}$ be the vectors normal to the palnes $A B C$ and $A B D$ respectively.
$n_{1} = A B \times A C = - i + 2 j - k n_{2} = A B \times A D = i + j - 2 k$
Let $θ$ be the acute angle between the planes, then $θ$ is the acute angle between their normals $n_{1}$ and $n_{2}$
$∴ cos θ = \frac{| - 1 + 2 + 2 |}{\sqrt{6} . \sqrt{6}} = \frac{3}{2} = \frac{1}{2} = cos \frac{π}{3} \Rightarrow θ = \frac{π}{3}$

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