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

Three-Dimensional Geometry · Question 81 of 120

Question 81

HardOriginal question

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

\frac{\pi}{6}

Correct answerYour answer

\frac{\pi}{4}

Correct answerYour answer

\frac{\pi}{3}

Correct answerYour answer

\frac{2 \pi}{3}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

Only the directions matter: $(1)(2) + (1)(-1) + (2)(1) = 3$ , and each has length $\sqrt{6}$ .
$\cos \theta = \frac{3}{6} = \frac{1}{2}$ , so $\theta = \frac{\pi}{3}$ .

Worth noting

$\frac{\pi}{6}$ reads $\frac{1}{2}$ as a sine; the dot product gives the cosine.

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

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