Find the angle between the lines parallel to the following pairs of vectors $i + 2 j + 3 k, - i + j + 2 k .$

{cos}^{- 1} (\sqrt{\frac{5}{12}}) .

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{cos}^{- 1} (\sqrt{\frac{7}{12}}) .

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{cos}^{- 1} (\sqrt{\frac{5}{7}}) .

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{sin}^{- 1} (\sqrt{\frac{5}{12}}) .

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Solution

The correct option is B ${cos}^{- 1} (\sqrt{\frac{7}{12}}) .$
Angle between two lines
$cos θ = ∣ ∣ ∣ ∣ ∣ \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{{(a_{1})}^{2} + {(b_{1})}^{2} + {(c_{1})}^{2}} \sqrt{{(a_{2})}^{2} + {(b_{2})}^{2} + {(c_{2})}^{2}}} ∣ ∣ ∣ ∣ ∣$
$= ∣ ∣ ∣ \frac{- 1 + 2 + 6}{\sqrt{1 + 4 + 9} \sqrt{1 + 1 + 4}} ∣ ∣ ∣ = ∣ ∣ ∣ \frac{7}{\sqrt{14} \sqrt{6}} ∣ ∣ ∣ = \sqrt{\frac{7}{12}}$
$∴ θ = {cos}^{- 1} \sqrt{\frac{7}{12}}$

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