# How do you find the inner product and state whether the vectors are perpendicular given <3,5>*<4,-2>?

Nov 2, 2016

The inner product is $2$
The vectors are not perpendicular.

#### Explanation:

Inner Product Definition
If $\vec{u} = \left\langle\left({u}_{1} , {u}_{2}\right)\right\rangle$, and $\vec{v} = \left\langle\left({v}_{1} , {v}_{2}\right)\right\rangle$, then the inner product (or dot product), a scaler quantity, is given by:
$\vec{u} \cdot \vec{v} = {u}_{1} {v}_{1} + {u}_{2} {v}_{2}$

Inner Product = 0 $\Leftrightarrow$ vectors are perpendicular

So,
Let $\vec{A} = \left\langle3 , 5\right\rangle$, and $\vec{B} = \left\langle4 , - 2\right\rangle$

Then the inner product is given by;
$\vec{A} \cdot \vec{B} = \left(3\right) \left(4\right) + \left(5\right) \left(- 2\right)$
$\vec{A} \cdot \vec{B} = 12 - 10 = 2$

$\vec{A} \cdot \vec{B} \ne 0 \Rightarrow$ vectors are not perpendicular