What is the shortest distance between L₁ and L₂?

Question 113

HardOriginal question

Two lines are $L_1: \vec{r} = (\hat{i} + 2\hat{k}) + \lambda(\hat{i} + 2\hat{j} - \hat{k})$ and $L_2: \vec{r} = (\hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + \hat{k})$ .

What is the shortest distance between $L_1$ and $L_2$ ?

\frac{1}{9}

Correct answerYour answer

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

Correct answerYour answer

\sqrt{3}

Correct answerYour answer

3

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

$\lvert \vec{b_1} \times \vec{b_2} \rvert = \sqrt{9 + 9 + 9} = 3\sqrt{3}$ .
$d = \frac{3}{3\sqrt{3}} = \frac{1}{\sqrt{3}}$ .

Worth noting

$\frac{1}{9}$ divides by $\lvert \vec{b_1} \times \vec{b_2} \rvert^{2} = 27$ , not by its length.

Concept: The shortest distance between skew lines
Typical timeTypical: 150 s
Your timeYours: —

The rule this testsSkew lines and the shortest distance

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

Practise this chapter →

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

What is the shortest distance between L₁ and L₂?

Question 113

Explanation