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What is the nature of the intersection between the line (x-1)/1 = (y-2)/-2 = (z-1)/3 and the plane x + 2y + z = 6?

1. They do not intersect at any point
2. They intersect at exactly one point
3. They intersect in infinitely many points
4. None of the above

Correct answer: They intersect in infinitely many points

Solution

The line's direction (1,-2,3) is perpendicular to the plane's normal (1,2,1) (dot product 0), and the point (1,2,1) satisfies x+2y+z=6. So the line lies entirely in the plane, giving infinitely many intersection points.

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