# If A= <-5 ,3 ,-3 > and B= <2 ,2 ,-7 >, what is A*B -||A|| ||B||?

Jun 16, 2018

$\boldsymbol{\underline{A}} \cdot \boldsymbol{\underline{B}} - | | \boldsymbol{\underline{A}} | | \setminus | | \boldsymbol{\underline{B}} | | = 17 - \sqrt{43} \sqrt{57}$

#### Explanation:

We have:

$\boldsymbol{\underline{A}} = \left\langle- 5 , 3 , - 3\right\rangle$ and $\boldsymbol{\underline{B}} = \left\langle2 , 2 , - 7\right\rangle$

And so we compute the Scalar (or dot product):

$\boldsymbol{\underline{A}} \cdot \boldsymbol{\underline{B}} = \left\langle- 5 , 3 , - 3\right\rangle \cdot \left\langle2 , 2 , - 7\right\rangle$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = \left(- 5\right) \left(2\right) + \left(3\right) \left(6\right) + \left(- 3\right) \left(- 7\right)$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = - 10 + 6 + 21$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = 17$

And we compute the vector norms (or magnitudes):

$| | \boldsymbol{\underline{A}} | | = | | \left\langle- 5 , 3 , - 3\right\rangle | |$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{\left\langle- 5 , 3 , - 3\right\rangle \cdot \left\langle- 5 , 3 , - 3\right\rangle}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{{\left(- 5\right)}^{2} + {\left(3\right)}^{2} + {\left(- 3\right)}^{2}}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{25 + 9 + 9}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{43}$

Similarly,

$| | \boldsymbol{\underline{B}} | | = | | \left\langle2 , 2 , - 7\right\rangle | |$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{\left\langle2 , 2 , - 7\right\rangle \cdot \left\langle2 , 2 , - 7\right\rangle}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{{\left(2\right)}^{2} + {\left(2\right)}^{2} + {\left(- 7\right)}^{2}}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{4 + 4 + 49}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{57}$

So that:

$\boldsymbol{\underline{A}} \cdot \boldsymbol{\underline{B}} - | | \boldsymbol{\underline{A}} | | \setminus | | \boldsymbol{\underline{B}} | | = 17 - \sqrt{43} \sqrt{57}$