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Flip to reveal answersWhat two things fix a plane in space?
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All 8 Flashcards — Equation of a plane
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Question
What two things fix a plane in space?
Answer
One point on the plane plus a normal vector n (a direction perpendicular to the plane).
Question
What is the scalar-product (vector) form of a plane?
Answer
r·n = a·n, where n is the normal and a is the position vector of a known point on the plane.
Question
What is the Cartesian form of a plane?
Answer
ax + by + cz = d, where (a, b, c) is the normal n and d = a·n.
Question
How do you read the normal off a Cartesian plane equation?
Answer
The coefficients of x, y, z are the components of the normal: ax + by + cz = d → n = (a, b, c).
Question
How do you find the constant d for a plane?
Answer
Substitute a known point on the plane into ax + by + cz; the value you get is d (which equals a·n).
Question
How do you check if a point lies on a plane?
Answer
Substitute the point's coordinates into the equation; if the left-hand side equals the right-hand side, the point is on the plane.
Question
Plane through (1, 2, −1) with normal (3, −1, 2): scalar-product form?
Answer
r·(3, −1, 2) = (1)(3)+(2)(−1)+(−1)(2) = −1, so r·(3, −1, 2) = −1.
Question
Is (2, 6, −4) a valid normal for the plane x + 3y − 2z = 7?
Answer
Yes — it is 2×(1, 3, −2), and any non-zero scalar multiple of the normal is still a normal.
Read the notes
Full study notes for Equation of a plane
Topic 3.17 hub
Vector planes (HL only)
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