Find an orthonormal basis of a plane, given the plane.?

Question

Given the plane x1 -2x2 - x3=0 find an orthonormal basis. How would I go about doing this? Please step by step. I'm reviewing material that I covered last year and it is puzzling me. Thanks in advance.

Northstar · Accepted Answer

Find two vectors u and v, that lie in the plane that are perpendicular to each other. Then normalize them. Both vectors will be orthogonal to the normal vector n, of the plane. And the dot product of orthogonal vectors is zero.

n = <1, -2, -1>

Let

u = <1, 0, 1>

Vector v is orthogonal to both n and u. Take the cross product.

v = n X u = <1, -2, -1> X <1, 0, 1> = <-2, -2, 2>

Any non-zero multiple of v will also be orthogonal to both vectors. Divide by -2.

v = <1, 1, -1>

Divide the vectors u and v by their magnitude.

u = <1, 0, 1>/√2 = <1/√2, 0, 1/√2>

v = <1, 1, -1>/√3 = <1/√3, 1/√3, -1/√3>

gack · Answer

A airplane is a 2-dimensional area, so a foundation of this airplane is going to be 2 vectors, not 3. carry a sheet of paper in front of you at an arbitrary attitude. you try to hit upon a foundation for that sheet of paper. you need to draw an x and y axis on it, yet not a z-axis. A vector which lies in this airplane could fulfill x1 + x2 + x3 = 0. in case you attempt your pronounced vectors, working example [a million, 0, 0], you will discover that x1 + x2 + x3 = a million + 0 + 0 = a million. on condition that x1 + x2 + x3 isn't 0, this does not lie interior the specified airplane. A vector that lies in this airplane is [a million, 0, -a million] by using fact x1 + x2 + x3 = a million + 0 + (-a million) = 0.

Find an orthonormal basis of a plane, given the plane.?

2 Answers