Vector Review

$a = [ a_1, a_2, \ldots, a_n ]^T$

$b = [ b_1, b_2, \ldots, b_n ]^T$

$a \cdot b = \langle a, b \rangle = \sum_i a_i b_i$

$(x,y) \cdot e_1 = 1 x + 0 y = x$

$(x,y) \cdot e_2 = 0 x + 1 y = y$

This relies on $|e_1| = |e_2| = 1$.

Axioms of the dot product

$c u \cdot v = c (u \cdot v )$
$u \cdot v = v \cdot u$ (only for real numbers). For complex numbers: $u \cdot v = \overline{v \cdot u}$.
$u \neq 0 \implies u \cdot u > 0$
$(u + v) \cdot w = u \cdot w + v \cdot w$

The dot product is invariant under rotation

For rotation $R$, let $u' = R u$ and $v' = R v$:

Note that the dot product $a \cdot b$ is the same as the matrix product $a^\mathsf{T} b$.

$\begin{array}[t]{rl} u' \cdot v' & = R u \cdot R v \\ & = (R u)^\mathsf{T} (R v) \\ & = (u^\mathsf{T} R^\mathsf{T}) (R v) \\ & = u^\mathsf{T} (R^\mathsf{T} R) v \\ & = u^\mathsf{T} v \\ & = u \cdot v \end{array}$.

This works because $R$ is a rotation matrix and $R^\mathsf{T} = R^{-1}$.

Geometric interpretation of dot product

$v \cdot \frac{u}{|u|}$ is the length of the projection of $v$ onto the line of $u$.
$(v \cdot \frac{u}{|u|}) \frac{u}{|u|}$ is the vector in the direction of $u$ with length equal to this projection.
Note that $|u| |u| = u \cdot u$.
Therefore, ${v \cdot u \over u \cdot u} u$ is the vector that is the projection of $v$ onto the line of $u$, and is the closest we can get to $v$ using only a multiple of $u$.
If $\theta$ is the angle between $u$ and $v$, then
$\begin{array}{rl} \cos \theta & = {\text{adjacent} \over \text{hypotenuse}} \\ & = {v \cdot \frac{u}{|u|} \over |v|} \end{array}$

So $u \cdot v = |u| |v| \cos\theta$.

Points vs. Vectors

Dot Product

Axioms of the dot product

The dot product is invariant under rotation

Geometric interpretation of dot product