Inner product, length, and orthogonality - Orthogonality and Least Squares

Notes:

Let vectors $u$ and $v$ be:

Then the inner product of the two vectors will be:

Let $u,v$ and $w$ be vectors in $\Bbb{R}^n$, and let $c$ be a scalar. Then,

1) $u \cdot v=v \cdot u$

2) $w(u+v)=w \cdot u+w \cdot v$

3) $(cu) \cdot v=c(u \cdot v)=u \cdot (cv)$

4) $u \cdot u \geq 0$, and $u \cdot u=0$ only if $u=0$

Suppose . Then the length (or norm) of a vector $v$ is

$\lVert v \rVert = \sqrt{v_{1}^2+v_{2}^2+\cdots +v_{n}^2}\;,\; \lVert v \rVert ^2 = v \cdot v$
Suppose dist$(u,v)$ is the distance between the vectors $u$ and $v$. To find the distance between the two vectors, we calculate
dist$(u,v)=\lVert u-v \rVert$
If two vectors $u$ and $v$ are orthogonal to each other, then it must be true that
$u \cdot v =0$