For which λ is PM perpendicular to L?

Question 51

MediumOriginal question

A point $P(5, 1, 4)$ and the line $L: \frac{x - 1}{2} = \frac{y}{2} = \frac{z - 3}{-1}$ are given. $M$ is the point of $L$ with parameter $\lambda$ , and $F$ is the foot of the perpendicular from $P$ to $L$ .

For which $\lambda$ is $PM$ perpendicular to $L$ ?

-1

Correct answerYour answer

1

Correct answerYour answer

\frac{11}{7}

Correct answerYour answer

9

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

$PM$ has direction ratios $2\lambda - 4, 2\lambda - 1, -1 - \lambda$ , and it must be perpendicular to $2, 2, -1$ .
$2(2\lambda - 4) + 2(2\lambda - 1) - (-1 - \lambda) = 9\lambda - 9 = 0$ , so $\lambda = 1$ .

Worth noting

9 is $(\vec{p} - \vec{a}) \cdot \vec{b}$ , which still has to be divided by $\lvert \vec{b} \rvert^{2} = 9$ .

Concept: The foot of the perpendicular from a point to a line
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsThe equation of a line

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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.

For which λ is PM perpendicular to L?

Question 51

Explanation