JEE Advanced · Mathematics · 30. Vector Algebra

Let two non-collinear unit vectors $\mathbf{a}$ and $\hat{\mathbf{b}}$ form an acute angle.
A point $P$ moves so that at any time $t$ the position vector $\mathbf{O P}$ (where, $O$ is the origin) is given by $\mathbf{a} \cos t+\hat{\mathbf{b}} \sin t$. When $P$ is farthest from origin $O$, let $M$ be the length of $\mathbf{O P}$ and $\hat{\mathbf{u}}$ be the unit vector along $\mathbf{O P}$. Then,

A
$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|}$ and $M=(1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$
B
$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}-\hat{\mathbf{b}}}{|\hat{\mathbf{a}}-\hat{\mathbf{b}}|}$ and $M=(1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$
C
$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|}$ and $M=(1+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$
D
$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}-\hat{\mathbf{b}}}{|\mathbf{\mathbf { a }}-\hat{\mathbf{b}}|}$ and $M=(1+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$

Verified Solution

Answer & Solution

Correct Answer

(A)
$\hat{\mathbf{u}}=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|}$ and $M=(1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}})^{1 / 2}$

Step-by-step Solution

Detailed explanation

$\mathbf{O P}=\hat{\mathbf{a}} \cos t+\hat{\mathbf{b}} \sin t$
\[
\begin{aligned}
& \Rightarrow \quad|\mathbf{O P}|=\sqrt{\left(\hat{\mathbf{a}} \cdot \hat{\mathbf{a}} \cos ^2 t+\hat{\mathbf{b}} \cdot \hat{\mathbf{b}} \sin ^2 t+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}} \sin t \cos t\right)} \\
& \Rightarrow \quad|\mathbf{O P}|=\sqrt{1+2 \hat{\mathbf{a}} \cdot \hat{\mathbf{b}} \cdot \sin t \cos t} \\
& \Rightarrow \quad|\mathbf{O P}|=\sqrt{1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}} \cdot \sin 2 t} \\
& \Rightarrow \quad|\mathbf{O P}|_{\max }=\sqrt{1+\hat{\mathbf{a}} \cdot \hat{\mathbf{b}}} \text { at } \sin 2 t=1 \Rightarrow t=\frac{\pi}{4} \\
& \Rightarrow \mathbf{O P}\left(\text { at } t=\frac{\pi}{4}\right)=\frac{1}{\sqrt{2}}(\hat{\mathbf{a}}+\hat{\mathbf{b}}) \\
&
\end{aligned}
\]
$\therefore$ Unit vector along $\mathbf{O P}$ at $\left(t=\frac{\pi}{4}\right)=\frac{\hat{\mathbf{a}}+\hat{\mathbf{b}}}{|\hat{\mathbf{a}}+\hat{\mathbf{b}}|}$

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The correct option is:

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