# What is the dot product of <8,-1,4> and <9,1,5 >?

Dec 28, 2015

$91$

#### Explanation:

For any 2 vectors $v = \left({v}_{1} , {v}_{2} , \ldots . , {v}_{n}\right) \mathmr{and} w = \left({w}_{1} , {w}_{2} , \ldots , {w}_{n}\right)$ in a real or complex finite dimensional vector space, the Euclidean inner product (dot product) is defined as follows :

$v \cdot w = {v}_{1} {w}_{1} + {v}_{2} {w}_{2} + \ldots . . + {v}_{n} {w}_{n}$.

So in this particular case of 3-dimesnional vectors we get

$\left(8 , - 1 , 4\right) \cdot \left(9 , 1 , 5\right) = \left(8 \times 9\right) + \left(- 1 \times 1\right) + \left(4 \times 5\right)$

$= 72 - 1 + 20$

$= 91$.

(Note that we can also represent the inner product as the angle between the 2 vectors and the norms of the 2 vectors as follows :
$u \cdot v = | | u | | \times | | v | | \cos \theta$