Angle Between a Line and a Plane
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Direct answer
sin θ = |b·n|/(|b||n|) gives the angle a line with direction b makes with a plane whose normal is n — sine, not cosine, because the angle is measured against the surface rather than the perpendicular. A line runs parallel to the plane exactly when b·n = 0, and strikes it head-on when b is a scalar multiple of n. The companions use cosines: between two planes, cos θ = |n1·n2|/(|n1||n2|); between two lines, cos θ = |b1·b2|/(|b1||b2|). Keeping the pairing straight is the entire difficulty; the arithmetic is one dot product between direction read from the symmetric form and normal read from the coefficients.
What you must remember
- The formula: sin θ = |b·n|/(|b||n|), with the modulus guarding the acute answer the options want.
- Read the direction free: the line (x − x1)/a = (y − y1)/b = (z − z1)/c hands you b = (a, b, c) with no work.
- Read the normal free: the plane ax + by + cz + d = 0 hands you n = (a, b, c) just as directly.
- Parallel condition: b·n = 0 — the line's direction is perpendicular to the plane's normal, hence inside the plane's directions.
- Perpendicular condition: b = kn for some scalar k, so a·a′ = b·b′ = c·c′ in ratios.
- Complement fact: the angle between the line and the normal is 90° − θ; if an option quotes cos⁻¹ of your dot-product value, that is the normal's angle, not the plane's.
- Plane-plane and line-line: both use cosine of dot products — only the line-plane pair switches to sine.
One line, one plane, one sine
Take the line (x − 1)/2 = (y + 2)/3 = (z − 4)/6 and the plane 3x + 2y + 6z = 7. The direction is b = (2, 3, 6) with |b| = √(4 + 9 + 36) = 7; the normal is n = (3, 2, 6) with |n| = √(9 + 4 + 36) = 7. The dot product is 2 × 3 + 3 × 2 + 6 × 6 = 6 + 6 + 36 = 48, so sin θ = 48/49 and θ = sin⁻¹(48/49) — about 78.5°, a line lying nearly flat inside the plane's directions? No: nearly perpendicular to the normal, so nearly parallel to the plane is wrong too — 78.5° from the plane's surface means steeply inclined, and the angle with the normal is the leftover 90° − 78.5° ≈ 11.5°, confirmed by cos φ = 48/49. This double reading is the drill: compute one dot product, then narrate both angles. Had the dot product come to zero, the line would sit parallel to the plane — and the follow-up question would be whether some point of it lies in the plane, decided by substituting (1, −2, 4): 3 − 4 + 24 = 23 ≠ 7, so it would hover strictly outside.
Frequently asked questions
What is the angle between a line and a plane?
sin θ = |b·n|/(|b||n|), where b is the line's direction and n the plane's normal — sine because the angle is measured with the surface.
When is a line parallel to a plane?
When b·n = 0, provided some point of the line does not satisfy the plane's equation; if it does, the line lies in the plane.
When is a line perpendicular to a plane?
When its direction is proportional to the normal: (a, b, c) = k(a′, b′, c′).
How does the line-plane formula differ from the plane-plane one?
Plane-plane and line-line angles use cosine of dot products; only the line-plane angle uses sine, since it is the complement of the normal's angle.
What is the angle between the line and the normal?
90° − θ, computable directly as cos⁻¹(|b·n|/(|b||n|)) — the value the wrong options keep quoting.