Zero Vector is Orthogonal to All Vectors

Theorem
Let $\mathbf V$ be a vector space of $n$ dimensions.

Let $\bszero \in \mathbf V$ be the zero vector.

Let $\mathbf a \in \mathbf V$ be an arbitrary vector in $\mathbf V$

Then $\bszero$ is orthogonal to $\mathbf a$.

Proof
By definition, $\mathbf a$ and $\mathbf b$ are orthogonal their dot product is zero:


 * $\mathbf a \cdot \mathbf b = 0$

We have:

where $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ is the standard ordered basis of $\mathbf V$.

By definition $1$ of dot product:


 * $\ds \mathbf a \cdot \mathbf b = \sum_{k \mathop = 1}^n a_k b_k$

By definition of zero vector:

Hence:

Hence the result.