Let two non-collinear vectors $\hat{a}$ and $\hat{b}$ form an acute angle. A point $\mathrm{P}$ moves, so…

Let two non-collinear vectors $\hat{a}$ and $\hat{b}$ form an acute angle. A point $\mathrm{P}$ moves, so that at any time $t$ the position vector $\overline{\mathrm{OP}}$, where $\mathrm{O}$ is origin, is given by $\hat{a} \sin t+\hat{b} \cos t$, when $P$ is farthest from origin $\mathrm{O}$, let $\mathrm{M}$ be the length of $\overline{\mathrm{OP}}$ and $\hat{\mathrm{u}}$ be the unit vector along $\overline{\mathrm{OP}}$, then
  1. $\hat{\mathrm{u}}=\frac{\hat{\mathrm{a}}+\hat{\mathrm{b}}}{|\hat{\mathrm{a}}+\hat{\mathrm{b}}|}$ and $\mathrm{M}=(1+\hat{\mathrm{a}} \cdot \hat{\mathrm{b}})^{\frac{1}{2}}$
  2. $\hat{\mathrm{u}}=\frac{\hat{\mathrm{a}}-\hat{\mathrm{b}}}{|\hat{\mathrm{a}}-\hat{\mathrm{b}}|}$ and $\mathrm{M}=(1+\hat{\mathrm{a}} \cdot \hat{\mathrm{b}})^{\frac{1}{2}}$
  3. $\hat{\mathrm{u}}=\frac{\hat{\mathrm{a}}+\hat{\mathrm{b}}}{|\hat{\mathrm{a}}+\hat{\mathrm{b}}|}$ and $\mathrm{M}=(1+\hat{\mathrm{2a}} \cdot \hat{\mathrm{b}})^{\frac{1}{2}}$
  4. $\hat{\mathrm{u}}=\frac{\hat{\mathrm{a}}-\hat{\mathrm{b}}}{|\hat{\mathrm{a}}-\hat{\mathrm{b}}|}$ and $\mathrm{M}=(1-\hat{\mathrm{2a}} \cdot \hat{\mathrm{b}})^{\frac{1}{2}}$

Solution

$\begin{aligned} & \mathrm{M}=|\overrightarrow{\mathrm{OP}}| \\ & M=\sqrt{(\hat{a} \sin t+\hat{b} \cos t)^2} \\ & =\sqrt{(\hat{a} \sin t)^2+(\hat{b} \cos t)^2+2(\hat{a} \sin t) \cdot(\hat{b} \cos t)} \\ & =\sqrt{\sin ^2 t+\cos ^2 t+\hat{a} \cdot \hat{b}(2 \sin t \cos t)} \\ & =\sqrt{1+\hat{\mathrm{a}} \cdot \hat{\mathrm{b}}(\sin 2 \mathrm{t})} \\ & \text { Maximum value of } \sin 2 t=1 \\ & \therefore \quad 2 \mathrm{t}=\sin ^{-1}(1) \\ & \therefore \quad \mathrm{t}=\frac{\pi}{4} \\ & \therefore \quad \mathrm{M}=\sqrt{1+\hat{\mathrm{a}} \cdot \hat{\mathrm{b}}(1)} \\ & =(1+\hat{a} \cdot \hat{b})^{\frac{1}{2}} \\ & \text { Now, } \hat{\mathrm{u}}=\frac{\overline{\mathrm{OP}}}{|\overline{\mathrm{OP}}|} \\ & =\frac{\hat{a} \sin t+\hat{b} \cos t}{|\hat{a} \sin t+\hat{b} \cos t|} \\ & =\frac{\hat{a}\left(\frac{1}{\sqrt{2}}\right)+\hat{b}\left(\frac{1}{\sqrt{2}}\right)}{\left|\hat{a}\left(\frac{1}{\sqrt{2}}\right)+\hat{b}\left(\frac{1}{\sqrt{2}}\right)\right|}\\ & \end{aligned}$ Unit vector of OP is $\hat{\mathrm{u}}=\frac{\hat{\mathrm{a}}+\hat{\mathrm{b}}}{|(\hat{\mathrm{a}}+\hat{\mathrm{b}})|}$

Asked in: MHT CET 2023 (09 May Shift 1)

Practice more Vectors questions on Aicharya