# If A= <6 ,-5 ,3 > and B= <5 ,4 ,-9 >, what is A*B -||A|| ||B||?

Mar 28, 2016

$A \cdot B - | A | \cdot | B | = - 17 - 92.4124 = - 109.4124$

#### Explanation:

Where $A \cdot B$ represents the dot product between vectors $A$ and $B$. This is given by:

$A \cdot B = \left({a}_{1} \cdot {b}_{1}\right) + \left({a}_{2} \cdot {b}_{2}\right) + \left({a}_{3} \cdot {b}_{3}\right)$.

Hence for $A = < 6 , - 5 , 3 >$ and $B = < 5 , 4 , - 9 >$

A*B=(6*5)+(-5)* 4+(3*(-9) or

$A \cdot B = 30 - 20 - 27 = - 17$

$| A |$ represents the magnitude of vector $A$ and is given by $\sqrt{{a}_{1}^{2} + {a}_{2}^{2} + {a}_{3}^{2}}$.

Hence $| A | \cdot | B |$

= $\left(\sqrt{{6}^{2} + {\left(- 5\right)}^{2} + {3}^{2}}\right) \cdot \left(\sqrt{{5}^{2} + {4}^{2} + {\left(- 9\right)}^{2}}\right)$

= $\left(\sqrt{36 + 25 + 9}\right) \cdot \left(\sqrt{25 + 16 + 81}\right)$

= $\sqrt{70} \cdot \sqrt{122} = 8.3666 \cdot 11.0454 = 92.4124$

Hence $A \cdot B - | A | \cdot | B | = - 17 - 92.4124 = - 109.4124$