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Member warned about not using the homework template

Given the non-zero vectors u, v and w in ℝ

Show that there is a non-zero linear combination of u and v that is orthogonal to w.

u and v must be linearly independant.

I am not really sure at all. But I have done this:

This is a screenshot of what I have done. Basicly, 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

^{3}Show that there is a non-zero linear combination of u and v that is orthogonal to w.

u and v must be linearly independant.

I am not really sure at all. But I have done this:

This is a screenshot of what I have done. Basicly, 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