# If A= <6 ,9 ,-1 > and B= <9 ,-1 ,8 >, what is A*B -||A|| ||B||?

Mar 23, 2016

$A \cdot B - | | A | | \cdot | | B | | = 37 - \sqrt{118} \cdot \sqrt{146}$

#### Explanation:

$A \cdot B$ is a dot product (or scalar product) and can be computed as follows:

$A = < \textcolor{b l u e}{6} , \textcolor{g r e e n}{\text{ } 9} , \textcolor{p u r p \le}{- 1} >$
$B = < \textcolor{b l u e}{9} , \textcolor{g r e e n}{- 1} , \textcolor{p u r p \le}{\text{ } 8} >$

$\implies \text{ } A \cdot B = \textcolor{b l u e}{6 \cdot 9} + \textcolor{g r e e n}{9 \cdot \left(- 1\right)} + \textcolor{p u r p \le}{\left(- 1\right) \cdot 8} = 37$

$| | A | |$ and $| | B | |$ can be computed as follows:

$| | A | | = \sqrt{{6}^{2} + {9}^{2} + {\left(- 1\right)}^{2}} = \sqrt{36 + 81 + 1} = \sqrt{118}$

$| | B | | = \sqrt{{9}^{2} + {\left(- 1\right)}^{2} + {8}^{2}} = \sqrt{81 + 1 + 64} = \sqrt{146}$

In total, you have

$A \cdot B - | | A | | \cdot | | B | | = 37 - \sqrt{118} \cdot \sqrt{146} \approx 94 , 26$