If $|\overline{\mathrm{a}}|=2,|\overline{\mathrm{b}}|=3,|\overline{\mathrm{c}}|=5$ and each of the angles…

If $|\overline{\mathrm{a}}|=2,|\overline{\mathrm{b}}|=3,|\overline{\mathrm{c}}|=5$ and each of the angles between the vectors $\bar{a}$ and $\bar{b}, \bar{b}$ and $\bar{c}$, $\overline{\mathrm{c}}$ and $\overline{\mathrm{a}}$ is $60^{\circ}$, then the value of $|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|$ is
  1. $\sqrt {69}$
  2. $\sqrt {70}$
  3. $\sqrt {80}$
  4. $\sqrt {39}$

Solution

$\begin{aligned} \overline{\mathrm{a} \cdot \overline{\mathrm{b}}} & =|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}| \cos 60^{\circ} \\ & =(2)(3)\left(\frac{1}{2}\right) \\ & =3 \\ \overline{\mathrm{b}} \cdot \overline{\mathrm{c}} & =|\overline{\mathrm{b}}||\overline{\mathrm{c}}| \cos 60^{\circ} \\ & =(3)(5)\left(\frac{1}{2}\right)=\frac{15}{2}\end{aligned}$ $\begin{aligned} \overline{\mathrm{a} \cdot \overline{\mathrm{c}}} & =|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{c}}| \cos 60^{\circ} \\ & =(2)(5)\left(\frac{1}{2}\right) \\ & =5 \\ \therefore \quad \mid \overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}^2 & =\left.\left|\overline{\mathrm{a}}^2+\right| \overrightarrow{\mathrm{b}}\right|^2+|\overline{\mathrm{c}}|^2+2(\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{a}}) \\ & =2^2+3^2+5^2+2\left(3+\frac{15}{2}+5\right) \\ & =4+9+25+31 \\ & =69 \\ \therefore \quad|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}| & =\sqrt{69}\end{aligned}$

Asked in: MHT CET 2023 (14 May Shift 2)

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