EXAMPLE $3$ Find the angle between the vectors $(4,2,-4$:$\}$ and $b=($:$4,-1,1$:$).$

SOLUTION Since

$|a|=\sqrt[\phantom{2}]{4^2+2^2+(-4{)}^2}=$

$$

and

$|b|=\sqrt[\phantom{2}]{4^2+(-1{)}^2+1^2}=$

$$

and since

we have from this corollary

$cos()=\frac{a*b}{|a||b|}=\frac{5\sqrt[\phantom{2}]{18}}{54}$

So the angle between a and $b$ is as follows. $($Round your final answer to two decimal places.$)$

$=co{s}^{-1}(\frac{5\sqrt[\phantom{2}]{18}}{54})$~~${?}_{\{\begin{array}{ccc}\text{:}[& \text{E}\text{n}\text{t}\text{e}\text{r}a& ]\\ & \text{n}\text{u}\text{m}\text{b}\text{e}\text{r}\text{.}& \end{array}}$