Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be vectors of magnitude 2,3…

Let $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be vectors of magnitude 2,3 and 4 respectively. If $\bar{a}$ is perpendicular to $(\bar{b}+\bar{c}), \bar{b}$ is perpendicular to $(\bar{c}+\bar{a})$ and $\vec{c}$ is perpendicular to $(\bar{a}+\bar{b})$, then the magnitude of $\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}$ is equal to
  1. 29
  2. $\sqrt{29}$
  3. 26
  4. $\sqrt{26}$

Solution

$\begin{aligned} & \overline{\mathrm{a}} \perp(\overline{\mathrm{b}}+\overline{\mathrm{c}}), \quad \overline{\mathrm{b}} \perp(\overline{\mathrm{c}}+\overline{\mathrm{a}}) \text { and } \overline{\mathrm{c}} \perp(\overline{\mathrm{a}}+\overline{\mathrm{b}}) \\ & \Rightarrow \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}+\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=0, \overline{\mathrm{~b}} \cdot \overline{\mathrm{c}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{a}}=0, \overline{\mathrm{c}} \cdot \overline{\mathrm{a}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{b}}=0 \\ & \Rightarrow \overline{\mathrm{a}} \cdot \overline{\mathrm{b}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{a}}=0\end{aligned}$ $\begin{aligned}|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|^2 & =|\overline{\mathrm{a}}|^2+|\overline{\mathrm{b}}|^2+|\overline{\mathrm{c}}|^2+2(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{a}}) \\ & =2^2+3^2+4^2+2(0) \\ & =4+9+16\end{aligned}$ $\begin{array}{ll}\therefore \quad & |\bar{a}+\bar{b}+\bar{c}|^2=29 \\ & \Rightarrow|\bar{a}+\bar{b}+\bar{c}|=\sqrt{29}\end{array}$

Asked in: MHT CET 2024 (09 May Shift 2)

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