# Find cartesian equation of hyperplane spanned by a set of vectors

Let W be a hyperplane in R4 spanned by the column vectors v1 , v2, and v3, where

Note that these are suppose to be COLUMN vectors:

v1 = [3,1, -2 , -1], v2 = [0, -1, 0 , 1], v3= [1,2 ,6, -2]

Find the Cartesian (i.e., linear) equation for W.

I'm not quite sure where to start or how to interpret this problem. I was thinking about first finding the span or column space?... But after that I would not know how to convert into a cartesian equation.

Any guidance or tips would be appreciated.

Thank you.

tiny-tim
Homework Helper
Good morning, Mr. Johnson, how are you! Hint: how would you find the Cartesian equation of the plane in R3 spanned by two column vectors v1 and v2 ? Ok so I found an example but I feel like it isn't very intuitive and that there are better methods and approaches to this problem.

So here is what I did:

The hyperplane W is of the form Ax1 + Bx2 + Cx3 + Dx4 = 0 since it must pass through the origin.

rref ( 3 1 -2 -1) = [1 0 0 0]
(0 -1 0 1 ) [ 0 1 0 -1]
(1 2 6 -2 ) [0 0 1 0]

Thus A = C =0
B = D

=> 0x1 + Bx2 +0x3 + Bx4 =0

Dividing by B => x2 + x4 =0 which is the final answer for the cartesian equation for W.

Can anyone verify? I know there is a better approach to this problem.

tiny-tim
Homework Helper
not actually following what you've done there ,

but your result x2 + x4 = 0 is obviously correct! the more general method i was thinking of (for two vectors in ℝ3) was to find their cross product using a determinant …

can you see a 4D equivalent of that? HallsofIvy