# If A= <2 ,-3 ,9 > and B= <0 , 3, 7 >, what is A*B -||A|| ||B||?

Jun 28, 2018

$\boldsymbol{\underline{A}} \cdot \boldsymbol{\underline{B}} - | | \boldsymbol{\underline{A}} | | \setminus | | \boldsymbol{\underline{B}} | | = 54 - \sqrt{94} \sqrt{58}$

#### Explanation:

We have:

$\boldsymbol{\underline{A}} = \left\langle2 , - 3 , 9\right\rangle$ and $\boldsymbol{\underline{B}} = \left\langle0 , 3 , 7\right\rangle$

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

$\boldsymbol{\underline{A}} \cdot \boldsymbol{\underline{B}} = \left\langle2 , - 3 , 9\right\rangle \cdot \left\langle0 , 3 , 7\right\rangle$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = \left(2\right) \left(0\right) + \left(- 3\right) \left(3\right) + \left(9\right) \left(7\right)$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = 0 - 9 + 63$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus \setminus = 54$

And we compute the vector norms (or magnitudes):

$| | \boldsymbol{\underline{A}} | | = | | \left\langle2 , - 3 , 9\right\rangle | |$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{\left\langle2 , - 3 , 9\right\rangle \cdot \left\langle2 , - 3 , 9\right\rangle}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{{\left(2\right)}^{2} + {\left(- 3\right)}^{2} + {\left(9\right)}^{2}}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{4 + 9 + 81}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{94}$

Similarly,

$| | \boldsymbol{\underline{B}} | | = | | \left\langle0 , 3 , 7\right\rangle | |$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{\left\langle0 , 3 , 7\right\rangle \cdot \left\langle0 , 3 , 7\right\rangle}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{{\left(0\right)}^{2} + {\left(3\right)}^{2} + {\left(7\right)}^{2}}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{0 + 9 + 49}$

$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = \sqrt{58}$

So that:

$\boldsymbol{\underline{A}} \cdot \boldsymbol{\underline{B}} - | | \boldsymbol{\underline{A}} | | \setminus | | \boldsymbol{\underline{B}} | | = 54 - \sqrt{94} \sqrt{58}$