The value of $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{k} \cdot(\hat{i} \times \hat{j})$ is

If $\overrightarrow{x}$ and $\overrightarrow{y}$ are two non-zero vectors such that $|\overrightarrow{x}+\overrightarrow{y}|=|\overrightarrow{x}|$ and $2\overrightarrow{x}+\lambda\overrightarrow{y}$ is perpendicular to $\overrightarrow{y}$,then the value of $\lambda$ is

Let $ABCD$ be a parallelogram such that $\vec{AB} = \vec{q}$ and $\vec{AD} = \vec{p}$,and $\angle BAD$ is an acute angle. If $\vec{r}$ is a vector that coincides with the altitude from vertex $B$ to the side $AD$,find $\vec{r}$.

The points $A(5, -1, 1)$,$B(7, -4, 7)$,$C(1, -6, 10)$,and $D(-1, -3, 4)$ are vertices of a:

Let $\vec{a} = -\hat{i} - \hat{k}$,$\vec{b} = -\hat{i} + \hat{j}$,and $\vec{c} = \hat{i} + 2\hat{j} + 3\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r} \times \vec{b} = \vec{c} \times \vec{b}$ and $\vec{r} \cdot \vec{a} = 0$,then the value of $\vec{r} \cdot \vec{b}$ is

$OA$ and $OB$ are two vectors of magnitudes $5$ and $6$ respectively. If $\angle BOA = 60^{\circ}$,then $OA \cdot OB$ is equal to