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Three-Dimensional Geometry · Question 52 of 120

What are the coordinates of F?

Question 52

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 $L$ .

What are the coordinates of $F$ ?

(19, 18, -6)

Correct answerYour answer

(-1, -2, 4)

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(3, 2, 2)

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(3, 2, 4)

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Show the answer and explanation

CorrectNot quiteAnswer C

Correct answer: C

Put $\lambda = 1$ in $M$ : $F = (1 + 2, 2, 3 - 1) = (3, 2, 2)$ .
Check: $PF$ has direction ratios $-2, 1, -2$ , and $2(-2) + 2(1) - 1(-2) = 0$ .

Worth noting

$(19, 18, -6)$ puts $\lambda = 9$ , the dot product before it is divided by $\lvert \vec{b} \rvert^{2}$ .

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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