Let $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}$. For what value of $\lambda$ is the vector $\vec{c} = \lambda\hat{i} + \hat{j} + (2\lambda - 1)\hat{k}$ parallel to the plane containing $\vec{a}$ and $\vec{b}$?

The number of integral values of $p$ for which the vectors $(p+1) \hat{i} - 3 \hat{j} + p \hat{k}$,$p \hat{i} + (p+1) \hat{j} - 3 \hat{k}$,and $-3 \hat{i} + p \hat{j} + (p+1) \hat{k}$ are linearly dependent is:

Consider the vectors $\vec{x}=\hat{i}+2\hat{j}+3\hat{k}$,$\vec{y}=2\hat{i}+3\hat{j}+\hat{k}$,and $\vec{z}=3\hat{i}+\hat{j}+2\hat{k}$. For two distinct positive real numbers $\alpha$ and $\beta$,define $\vec{X}=\alpha\vec{x}+\beta\vec{y}-\vec{z}$,$\vec{Y}=\alpha\vec{y}+\beta\vec{z}-\vec{x}$,and $\vec{Z}=\alpha\vec{z}+\beta\vec{x}-\vec{y}$. If the vectors $\vec{X}, \vec{Y}$,and $\vec{Z}$ lie in a plane,the value of $\alpha+\beta-3$ is $....$.

Let $\alpha \in \mathbb{R}$ and the three vectors $\vec{a} = \alpha \hat{i} + \hat{j} + 3\hat{k}$,$\vec{b} = 2\hat{i} + \hat{j} - \alpha \hat{k}$,and $\vec{c} = \alpha \hat{i} - 2\hat{j} + 3\hat{k}$. Then the set $S = \{ \alpha : \vec{a}, \vec{b}, \text{ and } \vec{c} \text{ are coplanar} \}$

If the vectors $a$,$b$,and $c$ are coplanar,then $\left|\begin{array}{ccc}a & b & c \\ a \cdot a & a \cdot b & a \cdot c \\ b \cdot a & b \cdot b & b \cdot c\end{array}\right|$ is equal to

If $[(\overline{a}+2 \overline{b}+3 \overline{c}) \times(\overline{b}+2 \overline{c}+3 \overline{a})] \cdot(\overline{c}+2 \overline{a}+3 \overline{b})=54$,then the value of $[\overline{a} \ \overline{b} \ \overline{c}]$ is