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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 $f(x)=x-\cos x$
$\therefore f ^{\prime}( x )=1+\sin x \Rightarrow f ^{\prime}( x ) \geq 0$
$\Rightarrow f ( x )$ is an increasing function and hence $f ( x )=0$ is true for a unique value of $x$.
As, $f\left(\frac{\pi}{6}\right)=\frac{\pi}{6}-\frac{\sqrt{3}}{2}<0$
and $f \left(\frac{\pi}{3}\right)=\frac{\pi}{3}-\frac{1}{2}>0$
So, using intermediate value theorem,
Thus, $\cos x=x$, for a unique value of $x, x \in\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$