Let $\vec{a}=\hat{i}$ and $\vec{b}=\hat{j}$. The point of intersection of the lines $\vec{r} \times \vec{a}=\vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b}=\vec{a} \times \vec{b}$ is:

Let $\overline{a}=3 \hat{i}-\alpha \hat{j}+\hat{k}$ and $\overline{b}=\hat{i}+\alpha \hat{j}+3 \hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\overline{a}$ and $\overline{b}$ is $8 \sqrt{3}$ sq. units,then $\overline{a} \cdot \overline{b}$ is equal to

If $\vec{a}, \vec{b}$ and $\vec{c}$ are vectors with magnitudes $2, 3$ and $4$ respectively, then the best upper bound of $|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2$ among the given values is

The least positive integral value of $\alpha$,for which the angle between the vectors $\alpha \hat{i}-2 \hat{j}+2 \hat{k}$ and $\alpha \hat{i}+2 \alpha \hat{j}-2 \hat{k}$ is acute,is

Coordinate planes and the planes $\pi_1, \pi_2, \pi_3$ which are respectively parallel to $YZ, ZX, XY$ planes at distances $a, b, c$,form a rectangular parallelepiped. $d_1$ is a diagonal of the face on the $XY$-plane not passing through the origin,and $d_2$ is a diagonal of plane $\pi_2$ coterminous with $d_1$. If none of the coordinates of the vertices of the parallelepiped are negative and the angle between $d_1$ and $d_2$ is $\theta$,then $\cos \theta=$

Let $\vec{a}$ be a vector in the plane containing vectors $\vec{b}=\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+\hat{k}$. If $\vec{a}$ is perpendicular to $\hat{i}+\hat{j}+3 \hat{k}$ and its projection on $\vec{b}$ is $3 \sqrt{6}$, then $|\vec{a}|^2=$