# If A= <7 , 2> and B= <-7, -1 >, what is ||A+B|| -||A|| -||B||?

Mar 19, 2016

$\textcolor{red}{|} | A + B | | - \textcolor{b l u e}{| | A | | - | | B | |} = \textcolor{red}{1} - \textcolor{b l u e}{.209} \approx .791$

#### Explanation:

Given:$A = < 7 , 2 >$ and $B = < - 7 , - 1 >$,
Required: $\textcolor{red}{|} | A + B | | - \textcolor{b l u e}{| | A | | - | | B | |}$
The red is the magnitude of the vector addition and the blue is
the sum of the magnitude of vectors.
$\textcolor{red}{R e d}$
$\vec{A} + \vec{B} = < 0 , 1 >$
and the magnitude is: $\textcolor{red}{| | A + B | | = 1}$

$\textcolor{b l u e}{B l u e}$
$| | A | | = \sqrt{{7}^{2} + {2}^{2}} = \sqrt{53}$
$| | A | | = \sqrt{{7}^{2} + {1}^{2}} = \sqrt{50}$
color(blue)(||A|| -||B||=sqrt(53)-sqrt(50)~~ .209

$\textcolor{red}{R e d} - \textcolor{b l u e}{B l u e} = \textcolor{red}{1} - \textcolor{b l u e}{.209} \approx .791$

Jul 1, 2018

||A+B|| - ||A|| - ||B|| = color(orange)(13.3512

#### Explanation:

$A = \left(\begin{matrix}7 \\ 2\end{matrix}\right)$

$B = \left(\begin{matrix}- 7 \\ - 1\end{matrix}\right)$

$A + B = \left(\begin{matrix}7 \\ 2\end{matrix}\right) + \left(\begin{matrix}- 7 \\ - 1\end{matrix}\right) = \left(\begin{matrix}0 \\ 1\end{matrix}\right)$

$| | A + B | | = \sqrt{{\left(0\right)}^{2} + {\left(1\right)}^{2}} = 1$

$| | A | | = \sqrt{{\left(7\right)}^{2} + {\left(2\right)}^{2}} = \sqrt{53}$

$| | B | | = \sqrt{{\left(- 7\right)}^{2} + {\left(- 1\right)}^{2}} = \sqrt{50}$

:.||A+B|| - ||A|| - ||B|| = color(orange)(1-sqrt(53)- sqrt(50) ~~ - 13.3512