# What is the dot product of <5,1,-7> and <8,4,9>?

The dot product of two three dimensional vectors is defined as

$a \cdot b = {a}_{x} \cdot {b}_{x} + {a}_{y} \cdot {b}_{y} + {a}_{z} \cdot {b}_{z}$

In our case $a = \left(5 , 1 , - 7\right)$ and $b = \left(8 , 4 , 9\right)$

hence

$a \cdot b = 5 \cdot 8 + 1 \cdot 4 - 7 \cdot 9 = - 19$

Jan 3, 2016

$- 19$

#### Explanation:

To do the dot product, you multiply together the corresponding components, and then sum the results. In general:
$\left({a}_{1} , {a}_{2} , {a}_{3}\right) \cdot \left({b}_{1} , {b}_{2} , {b}_{3}\right) = \left({a}_{1} {b}_{1} + {a}_{2} {b}_{2} + {a}_{3} {b}_{3}\right)$

$\left(5 \cdot 8\right) + \left(1 \cdot 4\right) + \left(- 7 \cdot 9\right)$
$= 40 + 4 - 63$
$= - 19$