# How do you compute the dot product for <5, 12>*<-3, 2>?

Nov 9, 2016

Please see the explanation of how to compute the dot-product for any two vectors of the same dimension and the answer to the given vectors: $< 5 , 12 > \cdot < - 3 , 2 > = 9$

#### Explanation:

For two dimensional vectors $< {a}_{x} , {a}_{y} > \mathmr{and} < {b}_{x} , {b}_{y} >$, the dot-product is:

$< {a}_{x} , {a}_{y} > \cdot < {b}_{x} , {b}_{y} > = \left({a}_{x}\right) \left({b}_{x}\right) + \left({a}_{y}\right) \left({b}_{y}\right)$

The dot-product is extendable to 3 dimensions and beyond:

For 3 dimensional vectors $< {a}_{x} , {a}_{y} , {a}_{z} > \mathmr{and} < {b}_{x} , {b}_{y} , {b}_{z} >$, the dot-product is:

$< {a}_{x} , {a}_{y} , {a}_{z} > \cdot < {b}_{x} , {b}_{y} , {b}_{z} > = \left({a}_{x}\right) \left({b}_{x}\right) + \left({a}_{y}\right) \left({b}_{y}\right) + \left({a}_{z}\right) \left({b}_{z}\right)$

For n dimensional vectors $< {a}_{1} , {a}_{2} , \ldots , {a}_{n} > \mathmr{and} < {b}_{1} , {b}_{2} , \ldots , {b}_{n} >$, the dot-product is:

$< {a}_{1} , {a}_{1} , \ldots , {a}_{n} > \cdot < {b}_{1} , {b}_{2} , \ldots , {b}_{n} > = \left({a}_{1}\right) \left({b}_{1}\right) + \left({a}_{2}\right) \left({b}_{2}\right) + \ldots , + \left({a}_{n}\right) \left({b}_{n}\right)$

The dot product for the given vectors is:

$< 5 , 12 > \cdot < - 3 , 2 > = \left(5\right) \left(- 3\right) + \left(12\right) \left(2\right)$

$< 5 , 12 > \cdot < - 3 , 2 > = - 15 + 24$

$< 5 , 12 > \cdot < - 3 , 2 > = 9$