Let $\vec{a}, \vec{b}, \vec{c}$ be co-initial vectors and $\vec{a}=2 \hat{i}-\hat{j}+5 \hat{k}$ and…

Let $\vec{a}, \vec{b}, \vec{c}$ be co-initial vectors and $\vec{a}=2 \hat{i}-\hat{j}+5 \hat{k}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}-\hat{\mathrm{k}}$. Let $(\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}})=\theta$ be an acute angle and $\overrightarrow{\mathrm{c}}$ be the vector along the bisector of the angle $\theta$. If $\lambda, x$, $\mathrm{y} \in \mathbb{R}$, then $\overrightarrow{\mathbf{c}}=$
  1. $\lambda(5 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})$
  2. $\lambda(-\hat{\mathrm{i}}-8 \hat{\mathrm{j}}+6 \hat{\mathrm{k}})$
  3. $(2 x+3 y) \hat{i}+(7 y-x) \hat{j}+(5 x-y) \hat{k}$
  4. $(2 x+3 y) \hat{i}+(x+7 y) \hat{j}+(5 x+y) \hat{k}$

Solution

$\because \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ Now, the angle bisector of vector $\vec{a} \& \vec{b}$ are $ \begin{aligned} & \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{a}}| \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{a}} \cdot|\overrightarrow{\mathrm{b}}| \\ & =\sqrt{30}(3 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}-\hat{\mathrm{k}})+(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}}) \cdot \sqrt{59} \end{aligned} +$ Let $\sqrt{30}=\mathrm{y} \& \sqrt{59}=\mathrm{x}$ then $ \begin{aligned} & \overrightarrow{\mathrm{c}}=(3 y \hat{i}+7 y \hat{j}-y \hat{k})+(2 x \hat{i}-x \hat{j}+5 x \hat{k}) \\ & \vec{c}=(3 y+2 x) \hat{i}(7 y-x) \hat{j}+(5 x-y) \hat{k} \end{aligned} $

Asked in: AP EAMCET 2023 (18 May Shift 2)

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