Which describes the lines r = (2 i - j + k) + λ( i + 2 j - 3 k) and (x - 4)/(-2) = (y - 3)/(-4) = (z + 5)/6?

Three-Dimensional Geometry · Question 104 of 120

Question 104

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Which describes the lines $\vec{r} = (2\hat{i} - \hat{j} + \hat{k}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})$ and $\frac{x - 4}{-2} = \frac{y - 3}{-4} = \frac{z + 5}{6}$ ?

Parallel and distinctCorrect answerYour answerThe same lineCorrect answerYour answerMeeting at one pointCorrect answerYour answerSkewCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

$-2, -4, 6 = -2(1, 2, -3)$ : the lines are parallel.
$\lambda = 2$ on the first line gives $(4, 3, -5)$ , a point of the second; sharing a point, they are the same line.

Worth noting

Parallel and distinct lines share no point; here $(4, 3, -5)$ lies on both.

Concept: Classifying a pair of lines
Typical timeTypical: 90 s
Your timeYours: —

The rule this testsSkew lines and the shortest distance

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Which describes the lines r⃗=(2i^−j^+k^)+λ(i^+2j^−3k^)\vec{r} = (2\hat{i} - \hat{j} + \hat{k}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})r=(2i^−j^​+k^)+λ(i^+2j^​−3k^) and x−4−2=y−3−4=z+56\frac{x - 4}{-2} = \frac{y - 3}{-4} = \frac{z + 5}{6}−2x−4​=−4y−3​=6z+5​?

Explanation

Which describes the lines $\vec{r} = (2\hat{i} - \hat{j} + \hat{k}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})$ and $\frac{x - 4}{-2} = \frac{y - 3}{-4} = \frac{z + 5}{6}$ ?