Linear combination and orthogonality

[ThomMathz]

Given the non-zero vectors u, v and w in ℝ³
Show that there is a non-zero linear combination of u and v that is orthogonal to w.
u and v must be linearly independent.

I am not really sure at all. But I have done this:
This is a screenshot of what I have done. basically, I assumed in the end that u and v are not orthogonal, and then I chose some suitable substitution for u and v, and ended up with zero when dotting them with w. Evverything is much appreciated and especially if you have another solution that is better or correct, because I think mine is not that good though.

https://gyazo.com/707e7c168a1c1a15166fd71be4ae7a81

 

[andrewkirk]

I'm pretty sure the claim is not correct. Consider the vectors
u=(5,0,0)
v=(5,1,0)
z=(5,0,1)

The set is linearly independent, but there is no linear combination of u and v that is orthogonal to w.

 

[Ray Vickson]

I'm pretty sure the claim is not correct. Consider the vectors
u=(5,0,0)
v=(5,1,0)
z=(5,0,1)

The set is linearly independent, but there is no linear combination of u and v that is orthogonal to w.

If by z you mean w, then the statement above is false: the linear combination c*u-c*v = (0,-c,0) is orthogonal to w = (5,0,1).

The result is true, but I will not look at the OP's screenshot (only at a typed version).