# If A= <-8 ,3 ,-4 > and B= <-3 ,4 ,8 >, what is A*B -||A|| ||B||?

$A \cdot B - | | A | | | | B | | = - 85$
Given vectors $\vec{u} = {u}_{x} \hat{i} + {u}_{y} \hat{j} + {u}_{z} \hat{k}$ and $\vec{v} = {v}_{x} \hat{i} + {v}_{y} \hat{j} + {v}_{z} \hat{k}$, $u \cdot v = {u}_{x} {v}_{x} + {u}_{y} {v}_{y} + {u}_{z} {v}_{z}$ and $| | u | | = \sqrt{{u}_{x}^{2} + {u}_{y}^{2} + {u}_{z}^{2}}$.
$A \cdot B = \left(- 8\right) \left(- 3\right) + \left(3\right) \left(4\right) + \left(- 4\right) \left(8\right) = 4$
$| | A | | = \sqrt{{\left(- 8\right)}^{2} + {\left(3\right)}^{2} + {\left(- 4\right)}^{2}} = \sqrt{89}$
$| | B | | = \sqrt{{\left(- 3\right)}^{2} + {\left(4\right)}^{2} + {\left(8\right)}^{2}} = \sqrt{89}$
$\therefore A \cdot B - | | A | | | | B | | = 4 - \sqrt{89} \sqrt{89} = 4 - 89 = - 85$