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

Mar 24, 2016

The dot (scalar) product $A \cdot B = 28$. The length of vector A, $| | A | | =$, and the length of vector B, $| | B | | =$. Over all, $A \cdot B - | | A | | | | B | | = 28 - 80.84 = - 52.84$ $u n i t s$.

#### Explanation:

The question is essentially 'What is the difference between the dot product of two vectors and the product of their lengths?'

First find the dot product:

 A*B = ((-8*-3)+(3*4)+(-1*8)
$= \left(24 + 12 - 8\right) = 28$ $u n i t s$

Now the length of each vector:

$| | A | | = \sqrt{{\left(- 8\right)}^{2} + {3}^{2} + {\left(- 1\right)}^{2}} = \sqrt{64 + 9 + 1} = \sqrt{74} \approx 8.6$

$| | B | | = \sqrt{{\left(- 3\right)}^{2} + {4}^{2} + {8}^{2}} = \sqrt{9 + 16 + 64} = \sqrt{89} \approx 9.4$

So the product $| | A | | | | B | | = 8.6 \times 9.4 = 80.84$