If $\vec{P} = (k, 2, 3)$ and $\vec{Q} = (0, 3, k)$ and $\vec{P} \perp \vec{Q}$,then what is the value of $k$?

Consider three vectors $A = \hat{i} + \hat{j} - 2\hat{k}$,$B = \hat{i} - \hat{j} + \hat{k}$,and $C = 2\hat{i} - 3\hat{j} + 4\hat{k}$. $A$ vector $X$ of the form $X = \alpha A + \beta B$ (where $\alpha$ and $\beta$ are scalars) is perpendicular to $C$. The ratio of $\alpha$ to $\beta$ is: (in $: 1$)

If the vectors $2\hat{i} + 2\hat{j} - 2\hat{k}$,$5\hat{i} + y\hat{j} + \hat{k}$,and $-\hat{i} + 2\hat{j} + 2\hat{k}$ are coplanar,then the value of $y$ is:

What happens to the direction and magnitude of a vector when it is multiplied by a positive and a negative scalar $\lambda$?

The edges of a parallelepiped are represented by the vectors $\hat{i} + 2\hat{j}$,$4\hat{j}$,and $\hat{j} + 3\hat{k}$. Find its volume.

If $|\vec{a}| = \sqrt{26}$,$|\vec{b}| = 7$,and $|\vec{a} \times \vec{b}| = 35$,find $\vec{a} \cdot \vec{b}$.