For what value of h is y in the plane?

Question

let v1=[1,2,-1]  v2=[-2,-1,1] and y=[4,-1,h]

 

For what value of h is y in the plane spanned by v1 and v2?

h=?

Answer 1

Dalia,

 

Independent, vectors V1 and V2 determine a plane.  If the vector Y is to be in that plane, then it has to be a linear combination of V1 & V2.   Thus for constants a & b, the following holds true:

 

aV1 + bV2 = Y,  or  a(1, 2, -1) + b(-2, -1, 1) = (4, -1, h) , so a separate equation for each component must hold:

 a -2b = 4

2a  -b   = -1

-a  +b  =   h , solve them for a, b and h: 3 equations & 3 unknowns.

 

If we add equations 2 & 3, then b drops out and  a = h - 1

Substitute this expression for a into equation #1, simplify and get b = (h - 5)/2

But h = (b - a) from equation 3, so get h = (h - 5)/2 - (h - 1)  =  -h/2  -3/2  so (3/2)h = -(3/2)  and h = +1

 

Hence Y = (4, -1, +1) is a vector that lies in the plane.