Find a basis for a given plane?

Question

2x-3y+4z=0

how do u do this?

Answer 1

As it is a plane in R^3, you know we need two linearly independent vectors in the plane to get a basis.

Easiest way is to pick a couple of easy points. For instance, let y = 0, z = 1, then x = -2, so (-2, 0, 1) is one vector in the plane. If we let z = 0, y = 2, then x = 3, so (3, 2, 0) is another vector in the plane. These are clearly linearly independent, so they form a basis for the plane.

Answer 2

A plane can be defined by a point and a normal vector to the plane. From the equation 2x - 3Y + 4Z = 0 the normal is readily apparent as [2,-3,4] and an easy point is (0,0,0).

This can be demonstrated from finding two vectors on a plane and performing a cross product. Cross products produce a normal vector to the original two vectors.