The shortest distance between the lines r = i + λ(2 i + 2 j + k) and r = ( i + p k) + μ(2 i - j - 2 k) is 2, where p > 0. What is p?

Question 119

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The shortest distance between the lines $\vec{r} = \hat{i} + \lambda(2\hat{i} + 2\hat{j} + \hat{k})$ and $\vec{r} = (\hat{i} + p\hat{k}) + \mu(2\hat{i} - \hat{j} - 2\hat{k})$ is 2, where $p > 0$ . What is $p$ ?

-3

Correct answerYour answer

\frac{1}{3}

Correct answerYour answer

3

Correct answerYour answer

27

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$\vec{b}_1 \times \vec{b}_2 = -3\hat{i} + 6\hat{j} - 6\hat{k}$ , of length 9, and $\vec{a}_2 - \vec{a}_1 = p\hat{k}$ .
$d = \frac{\lvert -6p \rvert}{9} = \frac{2p}{3} = 2$ , so $p = 3$ .

Worth noting

$\frac{1}{3}$ sets the numerator $6p$ equal to 2 without dividing by 9.

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