In a triangle $ABC$, $D$ and $E$ divide the sides $BC$ and $CA$ in the ratio $2:1$ respectively. If $P$ is the point of intersection of $AD$ and $BE$, then the ratio in which $P$ divides $AD$ is

If $\vec{a}$ and $\vec{b}$ are two unit vectors such that $\vec{a} + 2\vec{b}$ and $5\vec{a} - 4\vec{b}$ are perpendicular to each other,then the angle between $\vec{a}$ and $\vec{b}$ is ............. $^\circ$.

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{4}$. If $\theta$ is the angle between the vectors $(\hat{a}+\hat{b})$ and $(\hat{a}+2 \hat{b}+2(\hat{a} \times \hat{b}))$,then the value of $164 \cos ^{2} \theta$ is equal to.

If $\vec{a}, \vec{b}, \text{ and } \vec{c}$ are non-coplanar vectors,then the point of intersection of the line passing through the points $2 \vec{a}+3 \vec{b}-\vec{c}$ and $3 \vec{a}+4 \vec{b}-2 \vec{c}$ with the line joining the points $\vec{a}-2 \vec{b}+3 \vec{c}$ and $\vec{a}-6 \vec{b}+6 \vec{c}$ is

If $\vec{a}$ and $\vec{b}$ are unit vectors,then what is the angle between $\vec{a}$ and $\vec{b}$ for $\sqrt{3}\vec{a} - \vec{b}$ to be a unit vector (in $^{\circ}$)?

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