# How do you find the inner product and state whether the vectors are perpendicular given <4,9,-3>*<-6, 7,5>?

Oct 16, 2016

$< 4 , 9 , - 3 > \cdot < - 6 , 7 , 5 > = 24$

They are not perpendicular.

#### Explanation:

In ${\mathbb{R}}^{3}$, the inner product, or dot product, is given by

$< {x}_{1} , {y}_{1} , {z}_{1} > \cdot < {x}_{2} , {y}_{2} , {z}_{2} > = {x}_{1} {x}_{2} + {y}_{1} {y}_{2} + {z}_{1} {z}_{2}$

Two vectors are perpendicular if and only if their inner product is $0$.

In the case of our given vectors, then, we have

$< 4 , 9 , - 3 > \cdot < - 6 , 7 , 5 > = 4 \left(- 6\right) + 9 \left(7\right) + \left(- 3\right) \left(5\right)$

$= - 24 + 63 - 15$

$= 24$

As the inner product of the two vectors is not $0$, they are not perpendicular.