# If A= <-2 ,-5 ,3 > and B= <-7 ,4 ,-9 >, what is A*B -||A|| ||B||?

Apr 8, 2016

$A \cdot B - | | A | | | | B | | = - 107 , 48$

#### Explanation:

$A = < - 2 , - 5 , 3 >$
$B = < - 7 , 4 , - 9 >$

$\text{let's find ||A||}$

$| | A | | = \sqrt{{A}_{x}^{2} + {A}_{y}^{2} + {A}_{z}^{2}}$
$| | A | | = \sqrt{{\left(- 2\right)}^{2} + {\left(- 5\right)}^{2} + {3}^{2}}$
$| | A | | = \sqrt{4 + 25 + 9}$
$| | A | | = \sqrt{38}$

$\text{let's find ||B||}$

$| | B | | = \sqrt{{B}_{x}^{2} + {B}_{y}^{2} + {B}_{z}^{2}}$
$| | B | | = \sqrt{{\left(- 7\right)}^{2} + {4}^{2} + {\left(- 9\right)}^{2}}$
$| | B | | = \sqrt{49 + 16 + 81}$
$| | B | | = \sqrt{146}$

$\text{let's find } A \cdot B$

$A \cdot B = {A}_{x} \cdot {B}_{x} + {A}_{y} \cdot {B}_{y} + {A}_{z} \cdot {B}_{z}$

$A \cdot B = 14 - 20 - 27$
$A \cdot B = - 33$

$A \cdot B - | | A | | | | B | | = - 33 - \left(\sqrt{38} \cdot \sqrt{146}\right)$

$A \cdot B - | | A | | | | B | | = - 33 - \sqrt{5548}$
$A \cdot B - | | A | | | | B | | = - 33 - 74 , 48$
$A \cdot B - | | A | | | | B | | = - 107 , 48$