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Q.
The vectors $\vec{a}(x)=\cos x \hat{i}+\sin x \hat{j}$ and $\vec{b}(x)=x \hat{i}+\sin x \hat{j}$ are collinear for
Vector Algebra
Solution:
$\vec{a}(x)$ and $\vec{b}(x)$ are collinear if and only if $\cos x=x$.
Now let $f(x)=x-\cos x$, then $f'(x)=1+\sin x \geq 0$
$\Rightarrow f(x)$ is increasing and hence $f(x)=0$ for a unique value of $x$.
For $x \geq \frac{\pi}{3}, f(x)>\,0$ and $x<\,\frac{\pi}{6}, f(x)<\,0 .$ Thus
$\cos \,x=x$, for a unique value of $x, x \in\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$