What is the angle between L₁ and L₂?

Question 98

MediumOriginal 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 angle between $L_1$ and $L_2$ ?

\frac{\pi}{6}

Correct answerYour answer

\frac{\pi}{4}

Correct answerYour answer

\frac{\pi}{3}

Correct answerYour answer

\frac{\pi}{2}

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Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

$\vec{b_1} \cdot \vec{b_2} = 2 + 2 - 1 = 3$ , and each has length $\sqrt{6}$ .
$\cos \theta = \frac{3}{6} = \frac{1}{2}$ , so $\theta = \frac{\pi}{3}$ .

Worth noting

$\frac{\pi}{2}$ assumes skew lines are perpendicular; they can meet at any angle in direction.

Concept: The angle between two lines
Typical timeTypical: 120 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.

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

What is the angle between L₁ and L₂?

Question 98

Explanation