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How do you find where a line meets a plane?
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All Flashcards in Topic 3.18
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3.18.18 cards
How do you find where a line meets a plane?
Write the line's x, y, z in terms of λ, substitute into the plane's Cartesian equation, solve the resulting equation for λ, then put λ back into the line for the point.
After substituting, you solve for λ in which equation?
The plane's equation becomes one equation in λ; solve that.
Do you put λ back into the line or the plane to get the point?
Back into the LINE — that gives the (x, y, z) coordinates of the intersection.
What does it mean if substitution gives a false statement like 2 = 5?
The line is parallel to the plane and never meets it (no intersection).
What does it mean if substitution gives 0 = 0 (always true)?
Every λ works, so the line lies entirely in the plane.
When are the λ-terms guaranteed to cancel after substituting?
When the line's direction d is perpendicular to the plane's normal n, i.e. d·n = 0 (the line skims the plane).
Line r = (1,0,2)+λ(2,1,−1) and plane x+2y+z=9 — find the point.
x=1+2λ, y=λ, z=2−λ ⇒ (1+2λ)+2λ+(2−λ)=9 ⇒ 3λ+3=9 ⇒ λ=2 ⇒ (5, 2, 0).
If d·n = 0 but a point of the line does NOT satisfy the plane, the line is…
Parallel to the plane and outside it (misses it). If a point DID satisfy it, the line would lie in the plane.
3.18.28 cards
How do you find the DIRECTION of the line where two planes meet?
Take the cross product of the two normals: d = n₁ × n₂ (it lies in both planes).
How do you find a POINT on the line of intersection of two planes?
Fix one coordinate (often z = 0), then solve the two plane equations as a 2×2 system for the other two coordinates.
Formula for the angle between two planes?
cos θ = |n₁·n₂| / (|n₁||n₂|), using the planes' normals (absolute value gives the acute angle).
Formula for the angle between a line and a plane?
sin θ = |d·n| / (|d||n|) — SINE, because the angle is measured to the surface (90° from the normal).
Why does the line–plane angle use SINE but plane–plane uses COSINE?
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.
Two planes have perpendicular normals (n₁·n₂ = 0). What's the angle between the planes?
90° — the planes are perpendicular when their normals are.
Find the line of intersection of x+y+z=6 and x−y+2z=5.
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).
Why take the absolute value of the dot product in these angle formulas?
To report the ACUTE angle — without it a negative dot product would give the obtuse angle.
Topic 3.18 study notes
Full notes & explanations for Lines & planes (HL only)
Math AA exam skills
Paper structures, command terms & tips
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