If $a=\hat{i}+\hat{j}$ and $b=3 \hat{i}-2 \hat{j}$, then the vector $r$ satisfying the equations $r \times a=b \times a$ and $r \times b=a \times b$ 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 of a triangle with vertices $(1, 2, 0)$,$(1, 0, a)$,and $(0, 3, 1)$ is $\sqrt{6}$ sq. units. Then the values of '$a$' are:

The magnitude of the projection of the vector $2\hat{i}+\hat{j}+\hat{k}$ on the vector perpendicular to the plane containing the vectors $\hat{i}+\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}$ is:

For two given vectors $\bar{a}$ and $\bar{b}$,if the vectors $\overline{A}$ and $\overline{B}$ are such that $\overline{A}+\overline{B}=\bar{a}$,$\overline{A} \times \overline{B}=\bar{b}$,and $\overline{A} \cdot \bar{a}=1$,then $\overline{A}=$

Let $\hat{\alpha}, \hat{\beta}, \hat{\gamma}$ be three unit vectors such that $\hat{\alpha} \times (\hat{\beta} \times \hat{\gamma}) = \frac{1}{2}(\hat{\beta} + \hat{\gamma})$. If $\hat{\beta}$ is not parallel to $\hat{\gamma}$, then the angle between $\hat{\alpha}$ and $\hat{\beta}$ is