Spanning set of U={w€R3|w(dot)(3,-2,1)=0}

[daniel123244]

Homework Statement

Basically I am trying to find a spanning set for the plane through the origin with a normal of
(3,-2,1) that is an element of R3

"Let u = (3, 2, 1), and U ={wεR|w*u=0}"
Find a spanning set for U


2. The attempt at a solution

Just guessing blindly here:

(3,-2,1)=3(1,0,0)-2(0,1,0)+1(0,0,1)
we were taught to say (1,0,0)=e1, (0,1,0)=e2, and (0,0,1)=e3 so:

3e1-2e2+e3 would be a linear combination of this vector (?)

then if I let:
v=(x,y,z)=x(1,0,0)+y(0,1,0)+z(0,0,1)

u(dot)v=3xe1-2ye2+ze3=0

so does U=span{3xe1,-2ye2,ze3} ? I just don't know how to account for the fact that they equal 0 in the span.

Thank you to anyone who can help!

 

[lippyka]

3. The attempt at a solutionThe equation u • v = 0 is equivalent to 3x - 2y + z = 0. This equation represents a line in R3, the set of all vectors that are on this line form our spanning set U. Therefore, U = {(x, y, z)|3x - 2y + z = 0}.