# If A= <8 ,-5 ,6 > and B= <7 ,1 ,0 >, what is A*B -||A|| ||B||?

Mar 14, 2018

$\boldsymbol{\underline{A}} \cdot \boldsymbol{\underline{B}} - | | \boldsymbol{\underline{A}} | | \setminus | | \boldsymbol{\underline{B}} | | = 51 - 25 \sqrt{2}$

#### Explanation:

We start with the vectors $A$ and $B$

$\boldsymbol{\underline{A}} = \left\langle8 , - 5 , 6\right\rangle$
$\boldsymbol{\underline{B}} = \left\langle7 , 1 , 0\right\rangle$

The dot (or scalar) product is given by:

$\boldsymbol{\underline{A}} \cdot \boldsymbol{\underline{B}} = \left\langle8 , - 5 , 6\right\rangle \cdot \left\langle7 , 1 , 0\right\rangle$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = \left(8\right) \left(7\right) + \left(- 5\right) \left(1\right) + \left(6\right) \left(0\right)$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = 56 - 5 + 0$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = 51$

And the norms are given by:

$| | \boldsymbol{\underline{A}} | | = | | \left\langle8 , - 5 , 6\right\rangle | |$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{{8}^{2} + {\left(- 5\right)}^{2} + {6}^{2}}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{64 + 25 + 36}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{125}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = 5$

$| | \boldsymbol{\underline{B}} | | = | | \left\langle7 , 1 , 0\right\rangle | |$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{{7}^{2} + {1}^{2} + {0}^{2}}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{49 + 1 + 0}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{50}$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = 5 \sqrt{2}$

So then:

$\boldsymbol{\underline{A}} \cdot \boldsymbol{\underline{B}} - | | \boldsymbol{\underline{A}} | | \setminus | | \boldsymbol{\underline{B}} | | = 51 - 25 \sqrt{2}$