# If A= <3 ,1 ,-4 > and B= <4 ,-2 ,3 >, what is A*B -||A|| ||B||?

Feb 16, 2018

$\textcolor{b l u e}{- 29.46} \textcolor{w h i t e}{888}$ (2 .d.p)

#### Explanation:

$\boldsymbol{A} = \left[\begin{matrix}3 \\ 1 \\ - 4\end{matrix}\right]$ , $\textcolor{w h i t e}{88} \boldsymbol{B} = \left[\begin{matrix}4 \\ - 2 \\ 3\end{matrix}\right]$

$\boldsymbol{A \cdot B}$

This is the Dot Product and is defined as:

$\boldsymbol{A \cdot B} = | | \boldsymbol{A} | | \cdot | | \boldsymbol{B} | | \cdot \cos \theta$

The multiplication of $\boldsymbol{A \cdot B}$ is different from the normal way in which we multiply in algebra. Normally we multiply in the following way:

$\left(a + b + c\right) \left(d + e + f\right) = a d + a e + a f + b d + b e + b f + c d + c e + c f$

In the dot product we multiply as follows:

$\left(a + b + c\right) \left(d + e + f\right) = a d + b e + c f$

So we are just multiplying corresponding components and summing the results. This is where the alternative name Inner Product comes from.

For some vector $\boldsymbol{A} = \left[\begin{matrix}a \\ b \\ c\end{matrix}\right]$

$| | \boldsymbol{A} | |$ is the magnitude of the vector $\boldsymbol{A}$ and is defined as:

$| | \boldsymbol{A} | | = \sqrt{{a}^{2} + {b}^{2} + {c}^{2}}$

This is just the distance formula found in coordinate geometry.

To the example:

First we find the dot product of $\boldsymbol{A}$ and $\boldsymbol{B}$

$\boldsymbol{A \cdot B} = \left(3 \times 4\right) + \left(1 \times - 2\right) + \left(- 4 \times 3\right) = - 2$

Now we find the magnitudes of $\boldsymbol{A}$ and $\boldsymbol{B}$

$| | \boldsymbol{A} | | = \sqrt{{\left(3\right)}^{2} + {\left(1\right)}^{2} + {\left(- 4\right)}^{2}} = \sqrt{9 + 1 + 16} = \sqrt{26}$

$| | \boldsymbol{B} | | = \sqrt{{\left(4\right)}^{2} + {\left(- 2\right)}^{2} + {\left(3\right)}^{2}} = \sqrt{16 + 4 + 9} = \sqrt{29}$

Now we require:

$\boldsymbol{A \cdot B} - | | \boldsymbol{A} | | | | \boldsymbol{B} | |$

$- 2 - \sqrt{26} \sqrt{29} = \textcolor{b l u e}{- 29.46} \textcolor{w h i t e}{888}$ (2 .d.p)