Do two 2-D vectors that are not mutually perpendicular, constitute a pair of linearly independent vectors?

Question

Do two 2-D vectors that are not mutually perpendicular,  constitute a pair of linearly independent vectors?

irishboy123 · Answer

https://gyazo.com/2b28c690ecdbb50476c734cfb981c183 in brief, yes, unless they are parallel. if they are parallel $\vec u\times \vec v = 0$ so $$\left|\left|\begin{matrix}u_x & u_y \ v_x & v_y\end{matrix}\right|\right| = 0$$ as in this longer example: https://gyazo.com/a260ae04000052418719fb02795c2bf8

irishboy123 · Answer

oops longer example here https://gyazo.com/58587800e65fda229fbac26f34289a0a url appears in screengrab

anonymous · Answer

Thank you IrishBoy123.
But I want you to take just one pair of two 2_D vectors that are non ortogonal and show the proof.
You have taken a set of 3-D vectors. I am not interested in that.  I am interested only in the case where all the vectors are in one plane only. Only this case would be useful to me.
P. Radhakrishnamurty

sohailiftikhar · Answer

no they can't

anonymous · Answer

Dear sohailiftikhar.

How do we prove your statement?
P. Radhakrishnamurty