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

$u \cdot v =0$