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Answer on Linear Algebra Question for Muhammad Imran Qureshi

Question #7979
Find one vector in R^3 that spans the intersection of U and W where U is the xy-plane, U={(a,b,0)}, and W is the space spanned by the vectors (1,1,1) and (1,2,3).
Expert's answer
Let u=(a,b,c) be the vector spanning the intersection of U and W.

we find the equation of the plane W spanned by the vectors v=(1,1,1) and
The normal vector, n=(nx,ny,nz), to this plane is the
cross-product n=[v,w].
This vector has the following

nx = det 1 1 = 3-2 = 1
2 3

= det 1 1 = 1-3 = -2
3 1

nz = det 1 1 = 2-1 =
1 2

Thus n=(1,-2,1) and the equation of the plane

Thus u satisfies the equation:


Moreover, since u belongs to U, we have that c=0.
This gives
the relation
a-2b=0 => a=2b.

Hence we can take u=(2,1,0).

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