First slide

Straight lines in 3D

Question

The length of the perpendicular drawn from (1, 2,3) to the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ is

Moderate

A

4
B

5
C

6
D

7

Solution

Let P be the point (1,2,3) and PN be the length of the perpendicular from P on the given line.
Coordinates of point N are $(3 λ + 6, 2 λ + 7 - 2 λ + 7)$
Now PN is perpendicular to the given line or vector $3 \vec{i} + 2 \vec{j} - 2 \vec{k}$
$3 (3 λ + 6 - 1) + 2 (2 λ + 7 - 2) - 2 (- 2 λ + 7 - 3) = 0$
$λ = - 1$
Then, point N is (3, 5, 9)
PN= 1

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