Inner product, length, and orthogonality  Orthogonality and Least Squares
Inner product, length, and orthogonality
Lessons
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$(u,v)=\lVert uv \rVert$
If two vectors $u$ and $v$ are orthogonal to each other, then it must be true that
$u \cdot v =0$

Intro Lesson
Inner Product, Length, and Orthogonality Overview: