Practice question · Fill in the blanks
Complete the interpretation of a plane's coefficients.
In the plane equation ax + by + cz = d, the vector (a, b, c) is ______.
Word bank: a direction lying inside the plane · the normal vector, perpendicular to the plane · the position of a point on the plane · always the zero vector
Hints
- The equation comes from setting a dot product to zero, which vector is doing the dotting?
- Try z = 0: its coefficient vector is (0, 0, 1), which points straight up out of that plane.
Show the answer
In the plane equation ax + by + cz = d, the vector (a, b, c) is the normal vector, perpendicular to the plane.
Why
The coefficients ARE the normal, perpendicular to the plane. The equation is just the dot product condition n . (r - r0) = 0 written out. Mistaking the normal for a direction inside the plane inverts every parallel and perpendicular test.
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