# How do you find the inner product and state whether the vectors are perpendicular given <8,4>*<2,4>?

Jun 28, 2016

The inner product gives $32$, and so the vectors are not perpendicular.

#### Explanation:

The inner product of two vector (of equal length, of course), is simply given by the sum of the products of the coordinates with same index.

In generale, if you have two vectors $u = \left({u}_{1} , {u}_{2} , \ldots , {u}_{n}\right)$ and $v = \left({v}_{1} , {v}_{2} , \ldots , {v}_{n}\right)$, then the inner product $u \setminus \cdot v$ is given by

${u}_{1} {v}_{1} + {u}_{2} {v}_{2} + \ldots + {u}_{n} {v}_{n} = \setminus {\sum}_{i = 1}^{n} {u}_{i} {v}_{i}$.

Furthermore, two vectors are said to be perpendicular if their inner product is zero, i.e. $u \setminus \cdot v = 0$.

In your case, the inner product is

$8 \cdot 2 + 4 \cdot 4 = 16 + 16 = 32$

and so the vectors are not perpendicular.

Note: if your change the sign of any of the four coordinates, they actually become perpendicular: for example,

$\left(8 , 4\right) \setminus \cdot \left(2 , - 4\right) = 8 \cdot 2 - 4 \cdot 4 = 16 - 16 = 0$