# What is || <4, 5, -7 > - < 9, -2, 1 > ||?

Dec 28, 2016

$| < 4 , 5 , - 7 > - < 9 , - 2 , 1 > | = \sqrt{138}$

#### Explanation:

Given $\vec{a} = < 4 , 5 , - 7 >$ and $\vec{b} = < 9 , - 2 , 1 >$, we can subtract $\vec{b}$ from $\vec{a}$ by subtracting the components of $\vec{b}$ from those of $\vec{a}$.

$< 4 , 5 , - 7 > - < 9 , - 2 , 1 >$

$\implies < \left(4 - 9\right) , \left(5 - \left(- 2\right)\right) , \left(- 7 - 1\right) >$

$\implies < - 5 , 7 , - 8 >$

Let's call this new vector $\vec{c}$, where $\vec{c} = < {c}_{x} , {c}_{y} , {c}_{z} >$. The magnitude of $\vec{c}$ is then given by:

$| \vec{c} | = \sqrt{{\left({c}_{x}\right)}^{2} + {\left({c}_{y}\right)}^{2} + {\left({c}_{z}\right)}^{2}}$

$\implies | \vec{c} | = \sqrt{{\left(- 5\right)}^{2} + {\left(7\right)}^{2} + {\left(- 8\right)}^{2}}$

$\implies | \vec{c} | = \sqrt{25 + 49 + 64}$

$\implies | \vec{c} | = \sqrt{138}$

$\therefore | < 4 , 5 , - 7 > - < 9 , - 2 , 1 > | = \sqrt{138}$