What is the angle between the lines r = (3 i + j) + λ( i + 2 j + 2 k) and r = ( j - 2 k) + μ(3 i + 2 j + 6 k)?

Question 66

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What is the angle between the lines $\vec{r} = (3\hat{i} + \hat{j}) + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\vec{r} = (\hat{j} - 2\hat{k}) + \mu(3\hat{i} + 2\hat{j} + 6\hat{k})$ ?

\cos^{-1}\left(\frac{19}{49}\right)

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\cos^{-1}\left(\frac{19}{7}\right)

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\cos^{-1}\left(\frac{19}{3}\right)

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\cos^{-1}{\left(\frac{19}{21} \right)}

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Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

Only the directions matter: $\cos \theta = \frac{\lvert 3 + 4 + 12 \rvert}{3 \times 7}$ .
$\theta = \cos^{-1}\left(\frac{19}{21}\right)$ .

Worth noting

The points $3\hat{i} + \hat{j}$ and $\hat{j} - 2\hat{k}$ play no part in the angle.

Ten questions from Three-Dimensional Geometry, at your own pace.