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What two things fix a plane in space?
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All Flashcards in Topic 3.17
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3.17.18 cards
What two things fix a plane in space?
One point on the plane plus a normal vector n (a direction perpendicular to the plane).
What is the scalar-product (vector) form of a plane?
r·n = a·n, where n is the normal and a is the position vector of a known point on the plane.
What is the Cartesian form of a plane?
ax + by + cz = d, where (a, b, c) is the normal n and d = a·n.
How do you read the normal off a Cartesian plane equation?
The coefficients of x, y, z are the components of the normal: ax + by + cz = d → n = (a, b, c).
How do you find the constant d for a plane?
Substitute a known point on the plane into ax + by + cz; the value you get is d (which equals a·n).
How do you check if a point lies on a plane?
Substitute the point's coordinates into the equation; if the left-hand side equals the right-hand side, the point is on the plane.
Plane through (1, 2, −1) with normal (3, −1, 2): scalar-product form?
r·(3, −1, 2) = (1)(3)+(2)(−1)+(−1)(2) = −1, so r·(3, −1, 2) = −1.
Is (2, 6, −4) a valid normal for the plane x + 3y − 2z = 7?
Yes — it is 2×(1, 3, −2), and any non-zero scalar multiple of the normal is still a normal.
3.17.28 cards
How do you find the normal to the plane through three points A, B, C?
Form two in-plane vectors AB and AC, then take the cross product: n = AB × AC.
How do you find a plane containing a line and a point P?
Use the line's direction d and a vector AP from a point on the line to P; the normal is n = d × AP.
From parametric form r = a + λu + μv, how do you get a normal?
Cross the two in-plane direction vectors: n = u × v.
After finding the normal, how do you complete the plane's equation?
Write ax + by + cz = d using the normal as coefficients, then substitute a known point to find d.
Can you simplify the normal vector?
Yes — divide by any common factor (and the constant d by the same factor); it's still the same plane.
Plane through A(1,0,2), B(3,1,2), C(2,−1,4): the normal?
AB = (2,1,0), AC = (1,−1,2); AB × AC = (2, −4, −3).
How do you convert Cartesian 3x − 2y + z = 8 to scalar-product form?
Read the normal off the coefficients: r·(3, −2, 1) = 8.
How can you check a plane equation you've found is correct?
Substitute each given point — they should all satisfy the equation.
Topic 3.17 study notes
Full notes & explanations for Vector planes (HL only)
Math AA exam skills
Paper structures, command terms & tips
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