What is the distance between the parallel lines r = λ( i + 2 j + 2 k) and r = 3 i + μ( i + 2 j + 2 k)?

Question 109

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

\frac{2 \sqrt{2}}{3}

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2 \sqrt{2}

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3

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6 \sqrt{2}

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CorrectNot quiteAnswer B

Correct answer: B

$\vec{b} \times (\vec{a_2} - \vec{a_1}) = 6\hat{j} - 6\hat{k}$ , of length $6\sqrt{2}$ ; $\lvert \vec{b} \rvert = 3$ .
$d = \frac{6\sqrt{2}}{3} = 2\sqrt{2}$ .

Worth noting

3 is the distance between the two given points, measured along the $x$ -axis, not across the lines.

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