Equations of planes
Objective. Move between vector and Cartesian plane forms.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A plane in three dimensions can be described in vector form r = a + λb + μc (using a point and two direction vectors), or more compactly in Cartesian form ax+by+cz=d, where (a,b,c) is a normal vector to the plane. Converting between forms uses the vector (cross) product of the two direction vectors to find the normal, then substitutes a known point into the Cartesian equation to find d. For example, if a plane contains the point (1,0,2) and has normal (2,−1,3), its Cartesian equation is 2x−y+3z=2(1)−0+3(2)=8.
Worked example
A plane passes through the point (2,1,−1) and has normal vector (1,3,−2). Find the Cartesian equation of the plane.
- 1The Cartesian form is 1x+3y−2z=d for some constant d
- 2Substitute the known point (2,1,−1) to find d
- 3d = 1(2)+3(1)−2(−1) = 2+3+2
- 4d = 7, giving the equation x+3y−2z=7
Answer: x + 3y − 2z = 7
Concept mastery
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Practice
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Start practisingCommon mistake. Students use a direction vector lying in the plane as if it were the normal vector, giving a completely wrong Cartesian equation. The normal must be perpendicular to the plane, typically found via a cross product of two vectors within the plane.
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