If $|\bar{a}|=3,|\bar{b}|=4,|\bar{a}-\bar{b}|=5$, then $|\bar{a}+\bar{b}|=$

If $|\bar{a}|=3,|\bar{b}|=4,|\bar{a}-\bar{b}|=5$, then $|\bar{a}+\bar{b}|=$
  1. 9
  2. 25
  3. 5
  4. 4

Solution

$\begin{aligned} & |\overline{\mathrm{a}}+\overline{\mathrm{b}}|^2=|\overline{\mathrm{a}}-\overline{\mathrm{b}}|^2+4 \cdot \overline{\mathrm{a}} \cdot \overline{\mathrm{b}} \\ & \text { Now }|\overline{\mathrm{a}}-\overline{\mathrm{b}}|^2=|\overline{\mathrm{a}}|^2+|\overline{\mathrm{b}}|^2-2 \overline{\mathrm{a}} \cdot \overline{\mathrm{b}} \\ & \therefore(5)^2=(3)^2+(4)^2-2 \overline{\mathrm{a}} \cdot \overline{\mathrm{b}} \Rightarrow \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=0 \end{aligned}$ Substituting in (1), we get $|\overline{\mathrm{a}}+\overline{\mathrm{b}}|^2=|\overline{\mathrm{a}}-\overline{\mathrm{b}}|^2 \Rightarrow|\overline{\mathrm{a}}+\overline{\mathrm{b}}|=5$

Asked in: MHT CET 2021 (21 Sep Shift 1)

Practice more Vectors questions on Aicharya