Question

If $|\overrightarrow{a}|=5,|\overrightarrow{a}-\overrightarrow{b}|=8$ and $|\overrightarrow{a}+\overrightarrow{b}|=10$, then $|\overrightarrow{b}|$ is equal to:

• A. $1$
• B. $\sqrt{57}$
• C. $3$
• D. $57$

Answer 1

The correct option is B $\sqrt{57}$

Given,

$\left|\overline{a}\right|=5$

$\left|\overline{a}-\overline{b}\right|=8equation---(1$

$\left|\overline{a}+\overline{b}\right|=10equation---(2$

Squaring and adding both equation (1) & (2)

$\underset{–}{\begin{array}{c}{\left|a\right|}^2+{\left|b\right|}^2-2\left|a\right|\left|b\right|\cos \theta =64\\ {\left|a\right|}^2+{\left|b\right|}^2+2\left|a\right|\left|b\right|\cos \theta =100\end{array}}$

$\begin{array}{c}2\left({\left|a\right|}^2+{\left|b\right|}^2\right)=164\\ {\left|a\right|}^2+{\left|b\right|}^2=82\\ wehave,\\ \left|a\right|=5\\ then,\\ {\left(5\right)}^2+{\left|b\right|}^2=82\\ {\left|b\right|}^2=82-25\\ {\left|b\right|}^2=57\\ \left|b\right|=\sqrt{57}\end{array}$

So the correct Ans is option B.