Let $a=2 \hat{i}-3 \hat{j}+4 \hat{k}$, $b=7 \hat{i}+2 \hat{j}-3 \hat{k}$, and $c=\hat{i}+\hat{j}+\hat{k}$. The vector $x$ such that $x \cdot c=60$ and $x$ is perpendicular to both $a$ and $b$ is:

For any two vectors $a$ and $b$,if $a \times b = 0$,then

$A$ vector perpendicular to the plane containing the points $A(1, -1, 2)$,$B(2, 0, -1)$,and $C(0, 2, 1)$ 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:

The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8\hat{i} - 6\hat{j}$ and $3\hat{i} + 4\hat{j} - 12\hat{k}$ is:

Find a vector that is coplanar with $\hat{i} + \hat{j} + 2\hat{k}$ and $\hat{i} + 2\hat{j} + \hat{k}$ and perpendicular to $\hat{i} + \hat{j} + \hat{k}$.