Finding the equation of a plane
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Flip to reveal answersHow do you find the normal to the plane through three points A, B, C?
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All 8 Flashcards — Finding the equation of a plane
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Question
How do you find the normal to the plane through three points A, B, C?
Answer
Form two in-plane vectors AB and AC, then take the cross product: n = AB × AC.
Question
How do you find a plane containing a line and a point P?
Answer
Use the line's direction d and a vector AP from a point on the line to P; the normal is n = d × AP.
Question
From parametric form r = a + λu + μv, how do you get a normal?
Answer
Cross the two in-plane direction vectors: n = u × v.
Question
After finding the normal, how do you complete the plane's equation?
Answer
Write ax + by + cz = d using the normal as coefficients, then substitute a known point to find d.
Question
Can you simplify the normal vector?
Answer
Yes — divide by any common factor (and the constant d by the same factor); it's still the same plane.
Question
Plane through A(1,0,2), B(3,1,2), C(2,−1,4): the normal?
Answer
AB = (2,1,0), AC = (1,−1,2); AB × AC = (2, −4, −3).
Question
How do you convert Cartesian 3x − 2y + z = 8 to scalar-product form?
Answer
Read the normal off the coefficients: r·(3, −2, 1) = 8.
Question
How can you check a plane equation you've found is correct?
Answer
Substitute each given point — they should all satisfy the equation.
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Full study notes for Finding the equation of a plane
Topic 3.17 hub
Vector planes (HL only)
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