If $\bar{a}=\hat{i}+\hat{j}$ and $\bar{b}=2 \hat{i}-\hat{k}$,then the point of intersection of the lines $\bar{r} \times \bar{a}=\bar{b} \times \bar{a}$ and $\bar{r} \times \bar{b}=\bar{a} \times \bar{b}$ is

Let $ABC$ be a triangle of area $15 \sqrt{2}$ and the vectors $\overrightarrow{AB}=\hat{i}+2 \hat{j}-7 \hat{k}$,$\overrightarrow{BC}=a \hat{i}+b \hat{j}+c \hat{k}$ and $\overrightarrow{AC}=6 \hat{i}+d \hat{j}-2 \hat{k}$,where $d>0$. Then the square of the length of the largest side of the triangle $ABC$ is:

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}| = 3$ and $|\vec{b}| = \frac{\sqrt{2}}{3}$. For what angle $\theta$ between $\vec{a}$ and $\vec{b}$ is $\vec{a} \times \vec{b}$ a unit vector?

Two adjacent sides of a parallelogram are given by vectors $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$. Find the area of the parallelogram in square units.

The area of the parallelogram whose adjacent sides are $\vec{a} = \hat{i} - \hat{k}$ and $\vec{b} = 2\hat{j} + 3\hat{k}$ is

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$,$\vec{b} = 3\hat{i} - \hat{j} + 5\hat{k}$,and $\vec{c} = \hat{i} - 4\hat{j} - 2\hat{k}$ be three vectors. Let $\vec{r}$ be a vector perpendicular to both $\vec{b}$ and $\vec{c}$,and $\vec{r} \cdot \vec{a} = 11$. Then the vector among the following that is perpendicular to $\vec{r}$ is: