# If A= <2 ,6 ,-3 > and B= <3 ,-1 ,5 >, what is A*B -||A|| ||B||?

Jan 12, 2016

$15 - \sqrt{1715}$

#### Explanation:

If $A$ and $B$ are vectors, then $A . B = {\sum}_{i = 1}^{3} {x}_{a i} {y}_{b i}$ with ${a}_{i} , {b}_{i} \in \left\{1 , 2 , 3\right\}$.

$A . B = 2 \cdot 3 + 6 \cdot \left(- 1\right) + 5 \cdot \left(- 3\right) = 6 - 6 - 15 = 15$.

$| | A | | = \sqrt{{x}_{a}^{2} + {y}_{a}^{2} + {z}_{a}^{2}}$, so $| | A | | = \sqrt{{2}^{2} + {6}^{2} + {\left(- 3\right)}^{2}} = \sqrt{49}$ and $| | B | | = \sqrt{{3}^{2} + {\left(- 1\right)}^{2} + {5}^{2}} = \sqrt{35}$

Hence $A . B - | | A | | \cdot | | B | | = 15 - \sqrt{35 \cdot 49} = 15 - \sqrt{1715}$