Let $\vec{u} = \hat{i} + \hat{j}$,$\vec{v} = \hat{i} - \hat{j}$,and $\vec{w} = \hat{i} + 2\hat{j} + 3\hat{k}$. If $\hat{n}$ is a unit vector such that $\vec{u} \cdot \hat{n} = 0$ and $\vec{v} \cdot \hat{n} = 0$,then $|\vec{w} \cdot \hat{n}| = ....$

Let $\vec{a}=2 \hat{i}-3 \hat{j}-5 \hat{k}$ and $\vec{b}=3 \hat{i}+2 \hat{j}-5 \hat{k}$ be two vectors and $\vec{r}$ be a vector in the plane of $\vec{a}$ and $\vec{b}$. If $\vec{r}$ is orthogonal to the vector $5 \hat{i}-2 \hat{j}+3 \hat{k}$ and the magnitude of $\vec{r}$ is $\sqrt{94}$,then $|\vec{r} \cdot \vec{b}|=$

Let $\bar{a}=\bar{i}-\bar{j}+\bar{k}, \bar{b}=\bar{i}-2\bar{j}-2\bar{k}, \bar{c}=6\bar{i}+3\bar{j}-2\bar{k}$ be three vectors. If $\bar{d}$ is a vector perpendicular to both $\bar{a}$ and $\bar{b}$, and $|\bar{d} \times \bar{c}|=14$, then $|\bar{d} \cdot \bar{c}|=$

For any vector $x$, where $\hat{i}, \hat{j}, \hat{k}$ have their usual meanings, the value of $(x \times \hat{i})^{2} + (x \times \hat{j})^{2} + (x \times \hat{k})^{2}$ is equal to

Let $ABCD$ be a quadrilateral with $AB = a$, $AD = b$ and $AC = 3a + 2b$. If its area is $\alpha$ times the area of the parallelogram with $AB$ and $AD$ as adjacent sides, then the value of $\alpha$ is equal to

$(a - b) \times (a + b) = $