Zero Vector is Orthogonal to All Vectors

Theorem

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 $\mathbf a$.

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

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

We have:

$\ds \mathbf a$

$=$

$\ds \sum_{k \mathop = 1}^n a_k \mathbf e_k$

$\ds \mathbf b$

$=$

$\ds \sum_{k \mathop = 1}^n b_k \mathbf e_k$

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

By definition of dot product:

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

By definition of zero vector:

$\ds \bszero$

$=$

$\ds \sum_{k \mathop = 1}^n 0 \mathbf e_k$

Hence:

$\ds \bszero \cdot \mathbf a$

$=$

$\ds \sum_{k \mathop = 1}^n 0 \times a_k$

$\ds $

$=$

$\ds 0$

Hence the result.

$\blacksquare$