# If A= <8 ,3 ,-1 > and B= <6 ,4 ,3 >, what is A*B -||A|| ||B||?

Jul 1, 2018

Dot product is defined in ${\mathbb{R}}^{3}$ by

(a_1,a_2,a_3)·(b_1,b_2,b_3)=a_1b_1+a_2b_2+a_3b_3

By other hand vector module is defined by dot product by itself, it´s say sqrt(veca·veca) in the same terms of above.

In our case

vecA·vecB=8·6+3·4-1·3=57

$\left\mid \vec{A} \right\mid = \sqrt{{8}^{2} + {3}^{2} + {1}^{2}} = \sqrt{74}$

$\left\mid \vec{B} \right\mid = \sqrt{{6}^{2} + {4}^{2} + {3}^{2}} = \sqrt{61}$

Finally: vecA·vecB-abs(vecA)·abs(vecB)=57-sqrt74sqrt61=

$= 57 - 67.1863 = - 10.1863$