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Flip to reveal answersHow do you find the DIRECTION of the line where two planes meet?
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All 8 Flashcards — Planes meeting & angles
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Question
How do you find the DIRECTION of the line where two planes meet?
Answer
Take the cross product of the two normals: d = n₁ × n₂ (it lies in both planes).
Question
How do you find a POINT on the line of intersection of two planes?
Answer
Fix one coordinate (often z = 0), then solve the two plane equations as a 2×2 system for the other two coordinates.
Question
Formula for the angle between two planes?
Answer
cos θ = |n₁·n₂| / (|n₁||n₂|), using the planes' normals (absolute value gives the acute angle).
Question
Formula for the angle between a line and a plane?
Answer
sin θ = |d·n| / (|d||n|) — SINE, because the angle is measured to the surface (90° from the normal).
Question
Why does the line–plane angle use SINE but plane–plane uses COSINE?
Answer
The plane's normal is 90° to its surface, so the line-to-surface angle is the complement of the line-to-normal angle, swapping cos for sin.
Question
Two planes have perpendicular normals (n₁·n₂ = 0). What's the angle between the planes?
Answer
90° — the planes are perpendicular when their normals are.
Question
Find the line of intersection of x+y+z=6 and x−y+2z=5.
Answer
d = n₁×n₂ = (3,−1,−2); set z=0 ⇒ x=11/2, y=1/2. r = (11/2, 1/2, 0) + λ(3,−1,−2).
Question
Why take the absolute value of the dot product in these angle formulas?
Answer
To report the ACUTE angle — without it a negative dot product would give the obtuse angle.
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Full study notes for Planes meeting & angles
Topic 3.18 hub
Lines & planes (HL only)
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