Let Q(a, b, c) be the image of the point P(3, 2, 1) in the line and find the distance

Q. Let Q(a, b, c) be the image of the point P(3, 2, 1) in the line (x − 1)/1 = y/2 = (z − 1)/1. Then the distance of Q from the line (x − 9)/3 = (y − 9)/2 = (z − 5)/(−2) is

A. 8

B. 7

C. 6

D. 5

Correct Answer: 7

Explanation

The given line

(x − 1)/1 = y/2 = (z − 1)/1

passes through the point (1, 0, 1) and has direction vector

d = ⟨1, 2, 1⟩.

Let Q be the image of point P(3, 2, 1) in this line. The image of a point in a line lies such that the midpoint of P and Q lies on the line and PQ is perpendicular to the line.

Using the projection formula, coordinates of Q are obtained as:

Q = (−1, 6, −1)

Now, consider the second line:

(x − 9)/3 = (y − 9)/2 = (z − 5)/(−2)

A point on this line is A(9, 9, 5) and its direction vector is:

v = ⟨3, 2, −2⟩

Distance of point Q from a line is given by:

|AQ × v| / |v|

Substituting Q(−1, 6, −1) and A(9, 9, 5) and simplifying, the distance comes out to be:

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