Question #9a746

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Question #9a746

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Answer 1 · 2018-02-25T05:26:23

$(3, 4, 5)$ $(3,4,5)$

Explanation:

The simplest way of solving this is by using vectors.

The vector $- - \to P Q = (5, 4, 4) - (1, 4, 6) = (4, 0, - 2)$ $vec{PQ} = (5,4,4)-(1,4,6) = (4,0,-2)$ .

The foot of the perpendicular $R$ $R$ from $A$ $A$ is a point on $¯ ¯¯¯¯ ¯ P Q$ $bar{PQ}$ , so its coordinate vector is of the form
$m (5, 4, 4) + (1 - m) (1, 4, 6) = (1 + 4 m, 4, 6 - 2 m)$ $m(5,4,4)+(1-m)(1,4,6) = (1+4m,4,6-2m)$
so that the vector $- - \to A R$ $vec{AR}$ is

$- - \to A R = (1 + 4 m, 4, 6 - 2 m) - (1, 2, 1) = (4 m, 3, 5 - 2 m)$ $vec{AR} = (1+4m,4,6-2m)-(1,2,1)=(4m,3,5-2m)$

Since $- - \to A R$ $vec{AR}$ is perpendicular to $- - \to P Q$ $vec{PQ}$ , we have

$(4, 0, - 2) \cdot (4 m, 3, 5 - 2 m) = 0 \Rightarrow 20 m - 10 = 0 \Rightarrow m = \frac{1}{2}$ $(4,0,-2) cdot (4m,3,5-2m) = 0 implies 20m-10=0 implies m=1/2$

Thus, the coordinates of the foot of the perpendicular are
$(1 + 4 \times \frac{1}{2}, 4, 6 - 2 \times \frac{1}{2}) = (3, 4, 5)$ $(1+4times 1/2,4,6-2times 1/2) = (3,4,5)$

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Question #9a746

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