Why the zero vector always orthogonal to any other vector.

Question

Im currently looking at inner products and was wondering why the inner product of any vector with the zero vector is equal to 0. I have researched on this and only found the information that the zero vector is orthogonal to all vectors but no proof alongside.

And hence I was wondering if anyone had any proof as to why this happens.

Thanks in advance.

Edit:

Thanks every for their comment, I was now wondering if anyone could help explain to me why the inner product of $\left \langle M \vec u,\vec v \right \rangle$ = $\left \langle \vec u,M^{T} \vec v \right \rangle$

where M is and $n*n$ square matrix.

Answer 1

Let us denote the inner product by $( \cdot|\cdot).$

Then

$$(0|x)=(0x|x)=0(x|x)=0$$

for all $x$.

Answer 2

$$ \langle Mx,y\rangle =(Mx)^{T}y=x^{T}M^{T}y=x^{T}(M^{T}y)=\langle x,M^{T}y\rangle $$