For the line L: (x - 3)/6 = (y + 1)/(-2) = z/3, which of the following statements are true? I. L passes through (-3, 1, -3). II. The vector form of L is r = (3 i - j) + λ(6 i - 2 j + 3 k). III. L is parallel to 12 i + 4 j + 6 k. IV. The point of L with z = 6 is (15, -5, 6).

Question 53

MediumOriginal question

For the line $L: \frac{x - 3}{6} = \frac{y + 1}{-2} = \frac{z}{3}$ , which of the following statements are true?
I. $L$ passes through $(-3, 1, -3)$ .
II. The vector form of $L$ is $\vec{r} = (3\hat{i} - \hat{j}) + \lambda(6\hat{i} - 2\hat{j} + 3\hat{k})$ .
III. $L$ is parallel to $12\hat{i} + 4\hat{j} + 6\hat{k}$ .
IV. The point of $L$ with $z = 6$ is $(15, -5, 6)$ .

I, II and IV onlyCorrect answerYour answerI and II onlyCorrect answerYour answerAll four: I, II, III and IVCorrect answerYour answerII and IV onlyCorrect answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer A

Correct answer: A

$\lambda = -1$ gives $(-3, 1, -3)$ , so I is true; II reads the point $(3, -1, 0)$ and the ratios $6, -2, 3$ , so it is true.
$2(6\hat{i} - 2\hat{j} + 3\hat{k}) = 12\hat{i} - 4\hat{j} + 6\hat{k}$ , so III is false; $z = 3\lambda = 6$ gives $\lambda = 2$ and $(15, -5, 6)$ , so IV is true.

Worth noting

Leaving out IV misses that $\lambda = 2$ gives exactly $(15, -5, 6)$ .

Concept: The equation of a line through a point
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsThe equation of a line

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