A calculus of the absurd

Hopefully you know what perpendicular means. We would like to generalise this notion to a more abstract setting; we can say that

Definition 22.7.1 Let $\textsf {V}$ be an inner product space over a field $\textsf {F}$, then two vectors are orthogonal if and only if their inner product is zero; that is if we let $x, y \in \textsf {V}$
$\seteqnumber{0}{22.}{160}$
\begin{equation} \langle x, y \rangle = 0 \iff \text { $x$ and $y$ are orthogonal} \end{equation}

We can denote this as $x \bot y$ (read “$x$ and $y$ are orthogonal”).

This is not the most exciting definition, but it is a very useful one!

Example 22.7.1 Consider the vector space $\textsf {V} = \mathbb {R}^2$. The two vectors
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\begin{equation} \begin{pmatrix} 1 \\ 0 \end {pmatrix}, \begin{pmatrix} 0 \\ 1 \end {pmatrix} \end{equation}

are orthogonal with respect to the dot product.

We can just apply the definition, recall that

\begin{align} \begin{pmatrix} 1 \\ 0 \end {pmatrix} \cdot \begin{pmatrix} 0 \\ 1 \end {pmatrix} &= 1 \times 0 + 0 \times 1 \\ &= 0 \end{align}

And therefore, the two vectors are orthogonal.