Practice Planes in 3D in Math
Use these practice problems to test your method after reviewing the concept explanation and worked examples.
Quick Recap
A flat, infinite surface in three-dimensional space described by the equation ax + by + cz = d, where the vector \langle a, b, c \rangle is normal (perpendicular) to the plane.
Think of a plane as a perfectly flat, infinite floor that can be tilted at any angle in space. A horizontal floor is one plane; tilt it and you get another. To describe which tilt you have, imagine sticking a pole straight up out of the floor—that pole is the normal vector, and it captures the exact orientation of the surface. Any flat sheet in 3D, no matter how it's angled, is completely determined by where it sits and which direction its pole points.
Example 1
easyWrite the equation of the plane with normal vector \langle 2, -3, 1 \rangle passing through the point (1, 4, -2).
Example 2
mediumFind the equation of the plane through the points A(1, 0, 0), B(0, 2, 0), and C(0, 0, 3).
Example 3
mediumFind the distance from the point (3, -1, 2) to the plane 2x + y - 2z = 4.
Example 4
hardFind the angle between the planes x + 2y - z = 5 and 2x - y + 3z = 1.