Find the measure of the angle between the two lines whose direction ratios are $(2, 2, 1)$ & $(- 4, 1, 8)$ .

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Solution

Consider

$a_{1} = 2, b_{1} = 2, c_{1} = 1$

$a_{2} = - 4, b_{2} = 1, c_{2} 8$

And,

Let $(θ)$ be the angle between these two lines.

then

$\begin{matrix} cos θ = ∣ ∣ ∣ ∣ \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}} ∣ ∣ ∣ ∣ cos θ = ∣ ∣ ∣ \frac{- 8 + 2 + 8}{\sqrt{4 + 4 + 1} \sqrt{16 + 1 + 64}} ∣ ∣ ∣ cos θ = \frac{2}{\sqrt{9} \sqrt{81}} θ = {cos}^{- 1} (\frac{2}{27}) \end{matrix}$

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