What is a Non-zero Vector in R^3 that Belongs to Two Given Spans?

[AkilMAI]

Homework Statement

I require some help to find a non-zero vector in R^3 that belongs both to span {y; u} and to span {v;w} where y = (1; 0; 0); u = (0; 0; 1), v = (1; 1; 1) and w = (2; 3;-1),
I need to know if my below solutions is ok.Thank you

2. The attempt at a solution
Let 'a' be the required vector.
I need to satisfy [a,y,u]=0 and [a,v,w]=0; where [a,y,u] is the scalar triple product of a, y and u.
Since span of two given vectors is a plane, 'a' lies on the intersection of two planes hence 'a' is the vector along the line of intersection of the two planes.
=>...[A,B,C]=det(ABC)...so det(a,y,u)=>y=0...and det(a,v,w)=-4x+y+z=0...y=0,so x=1 and z=4

 

[Dick]

Right!

 

[AkilMAI]

ok:)...thanks