Help w/: Given u1 = (2,-1,0), u2 = (0,0,1), find w1, w2 such that (1,1,3) = w1+w2

[Belindaaaaaa]

Given u1 = (2,-1,0) and u2 = (0,0,1),

find w1 and w2 such that (1,1,3) = w1+w2,
where w1 lies in the plane passing through the origin, u1 and u2,
and w2 is perpendicular to the above mentioned plane.

I understand that u1 and u2 are orthogonal, but I don’t know how to continue solving the rest of the question. Please help, thank you!!

 

[JeffM]

Given u1 = (2,-1,0) and u2 = (0,0,1),

find w1 and w2 such that (1,1,3) = w1+w2,
where w1 lies in the plane passing through the origin, u1 and u2,
and w2 is perpendicular to the above mentioned plane.

I understand that u1 and u2 are orthogonal, but I don’t know how to continue solving the rest of the question. Please help, thank you!!

Two points are not orthogonal.

A good first step is to determine the function that describes the plane to which w_2 is perpendicular. Can you do that? What do you get? (Please show your work so we can catch errors.)

What methods have you been taught for finding perpendiculars to a plane?

 

[Belindaaaaaa]

Two points are not orthogonal.

A good first step is to determine the function that describes the plane to which w_2 is perpendicular. Can you do that? What do you get? (Please show your work so we can catch errors.)

What methods have you been taught for finding perpendiculars to a plane?

Why are the two points not orthogonal? The dot product of u1 and u2=0

I only managed to find the equation of the plane, which is -x+2y=0. Then I’m stuck

 

[pka]

Given u1 = (2,-1,0) and u2 = (0,0,1),
find w1 and w2 such that (1,1,3) = w1+w2,
where w1 lies in the plane passing through the origin, u1 and u2,
and w2 is perpendicular to the above mentioned plane.
I understand that u1 and u2 are orthogonal, but I don’t know how to continue solving the rest of the question. Please help, thank you!!

First \(\displaystyle u_1~\&~u_2\) as vectors are orthogonal. Moreover \(\displaystyle u_1\times u_2=<-1,-2,0>\) SEE HERE.
Thus the plane thru the origin and \(\displaystyle u_1~\&~u_2\) is \(\displaystyle x+2y=0\).