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Let N be the foot of the perpendicular dropped from the point P(1, 2, -1) onto the line L: x/1 = y/1 = z/(-1). A line is drawn from P, parallel to the plane x + y + 2z = 0, which intersects L at the point Q. If alpha is the acute angle between lines PN and PQ, then cos(alpha) equals:

1. 1/sqrt(5)
2. sqrt(3)/2
3. 1/sqrt(3)
4. 1/(2*sqrt(3))

Correct answer: 1/sqrt(5)

Solution

N is found by parameterising L as (t,t,-t) and minimising distance to P(1,2,-1). Q is another point on L satisfying the parallelism condition with the plane. The angle between PN and PQ is then computed via dot product.

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