Intersections: line-plane, plane-plane; angles practice questions.

The line r = (1, 0, 2) + λ(1, 1, 1) intersects the plane x + 2y - z = 5. Determine the point of intersection.

• (2, 1, 3)
• (4, 3, 5)
• (1, 0, 2)
• (5, 4, 6)

Answer: (4, 3, 5)

Substituting gives (1 + λ) + 2λ - (2 + λ) = 2λ - 1 = 5, so λ = 3 and the point is (4, 3, 5).

A line has direction vector d = (√3, 0, 1) and a plane has normal vector n = (0, 0, 1). Calculate the acute angle between the line and the plane.

• 30°
• 60°
• 45°
• 90°

Answer: 30°

sin θ = |d · n| / (|d||n|) = 1/2, so θ = 30°; using cos instead of sin gives the classic wrong answer 60°, which is the angle with the normal.

Calculate the acute angle between the planes x = 3 and x + y = 5.

• 30°
• 60°
• 90°
• 45°

Answer: 45°

The angle between planes equals the acute angle between their normals (1, 0, 0) and (1, 1, 0): cos θ = 1/√2, so θ = 45°.

The planes x + y + z = 6 and x - y + z = 2 intersect in a line. Which vector is parallel to that line?

• (1, 1, 1)
• (1, -1, 1)
• (1, 0, -1)
• (0, 1, 0)

Answer: (1, 0, -1)

The line of intersection is parallel to n₁ × n₂ = (1, 1, 1) × (1, -1, 1) = (2, 0, -2), i.e. parallel to (1, 0, -1); the normals themselves are not directions of the line.

Consider the line r = (0, 1, 1) + λ(2, 1, -1) and the plane x + 2z = 5. Determine the relationship between the line and the plane.

• The line is parallel to the plane, not meeting it
• The line lies entirely in the plane
• The line intersects the plane at exactly one point
• The line is perpendicular to the plane

Answer: The line is parallel to the plane, not meeting it

d · n = (2, 1, -1) · (1, 0, 2) = 2 - 2 = 0, so the line is parallel to the plane; the point (0, 1, 1) gives 0 + 2 = 2 ≠ 5, so it does not lie in the plane.

A line with direction vector (1, k, 2) is parallel to the plane 2x - y + 3z = 7. Determine the value of k.

• -8
• 8
• 4
• -4

Answer: 8

Parallel means d · n = 0: 2(1) - k + 3(2) = 8 - k = 0, so k = 8; adding instead of subtracting the middle term gives -8.