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Distance Formula

If θ is the a...

Question

If $θ$ is the angle between the lines $A B$ and $A C$ where $A, B, C$ are the three points with coordinates $(1, 2, - 1), (2, 0, 3), (3, - 1, 2)$ respectively, then $\sqrt{462} cos θ =$

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Solution

Given points : $A (1, 2, - 1), B (2, 0, 3), C (3, - 1, 2)$
$\Rightarrow - - \to A B =^i - 2^j + 4^k, - - \to A C = 2^i - 3^j + 3^k$
$\Rightarrow cos θ = \frac{- - \to A B \cdot - - \to A C}{| - - \to A B | | - - \to A C |} = \frac{2 + 6 + 12}{\sqrt{1 + 4 + 16} \sqrt{4 + 9 + 9}}$
$\Rightarrow cos θ = \frac{20}{\sqrt{21 \times 22}} = \frac{20}{\sqrt{462}} ∴ \sqrt{462} cos θ = 20$

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