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:

The value of $m$,if the vectors $\hat{\imath}-\hat{\jmath}-6 \hat{k}$,$\hat{\imath}-3 \hat{\jmath}+4 \hat{k}$,and $2 \hat{\imath}-5 \hat{\jmath}+m \hat{k}$ are coplanar,is

If $\overrightarrow{a} \cdot \overrightarrow{b} = 1, \overrightarrow{b} \cdot \overrightarrow{c} = 2$ and $\overrightarrow{c} \cdot \overrightarrow{a} = 3$,then the value of $[\vec{a} \times(\vec{b} \times \vec{c}), \vec{b} \times(\vec{c} \times \vec{a}), \vec{c} \times(\vec{b} \times \vec{a})]$ is

If the vectors $2i - 3j + 4k$, $i + 2j - k$ and $xi - j + 2k$ are coplanar, then $x = $

If $\alpha (a \times b) + \beta (b \times c) + \gamma (c \times a) = 0$ and at least one of the numbers $\alpha, \beta,$ and $\gamma$ is non-zero,then the vectors $a, b,$ and $c$ are

Let $\overrightarrow{A} = \hat{i} + \hat{j} + \hat{k}$,$\overrightarrow{B} = \hat{i}$,and $\overrightarrow{C} = C_1\hat{i} + C_2\hat{j} + C_3\hat{k}$. If $C_2 = -1$ and $C_3 = 1$,then to make the three vectors coplanar: