Let $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=2 \hat{j}-3 \hat{k}$. If $\vec{b}=\vec{c}-\vec{d}$, $\vec{a}$ is parallel to $\vec{c}$, and $\vec{a}$ is perpendicular to $\vec{d}$, then $\vec{c}+\vec{d}=$

If $a, b, c$ are unit vectors satisfying the relation $a+b+\sqrt{3} c=0$, then the angle between $a$ and $b$ is

In a triangle $ABC,$ if $|\overline{BC}|=8, |\overline{CA}|=7, |\overline{AB}|=10,$ then the projection of the vector $\overline{AB}$ on $\overline{AC}$ is equal to ....... .

If $\overline{e_1}, \overline{e_2}$ are two non-collinear unit vectors such that $|\overline{e_1}+\overline{e_2}|=\sqrt{3}$,then $(2 \overline{e_1}-5 \overline{e_2}) \cdot (3 \overline{e_1}+\overline{e_2}) = $

Let $ABCD$ be a parallelogram where $\overrightarrow{AB} = \overrightarrow{a}$,$\overrightarrow{AD} = \overrightarrow{b}$,$|\overrightarrow{a}| = |\overrightarrow{b}| = 2$ and $|\overrightarrow{a} \times \overrightarrow{b}| + \overrightarrow{a} \cdot \overrightarrow{b} = \sqrt{2} |\overrightarrow{a}| |\overrightarrow{b}|$,where $\overrightarrow{a} \cdot \overrightarrow{b} > 0$. Then the area of this parallelogram is (in square units):

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