[Math] Three vectors are coplanar then they are linearly dependent. Is it true for the bottom?

linear algebra

Suppose there are three vectors $\sin x,\cos x,\tan x$. Now I'm pretty sure that these lie on the same plane. But can I find any constant coefficients that satisfies $c_1\sin x + c_2\cos x + c_3\tan x =0$? I couldn't. Now please rectify me if there is some way to prove that they are linearly dependent. Or please explain if they aren't linearly dependent how they are lying on the same plane ($xy$ plane) .

Best Answer

Beware: strong confusion between vectors and functions!

Three coplanar vectors in $\mathbb{R}^3$ are indeed not linearly independent...

But the example you are giving is not three vectors, these are three functions... This is completely different.

And in that case, the three functions are clearly independent from each other, but these functions are not elements of $\mathbb{R}^3$

EDIT: side note: they are represented in the $xy$ plane, this is not to say they are elements of thus plane...

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