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 $\vec{b}$?

Let $a = i - k$, $b = xi + j + (1 - x)k$, and $c = yi + xj + (1 + x - y)k$. Then $[a\,b\,c]$ depends on

If the volume of the tetrahedron whose coterminous edges are given by the vectors $\vec{a} = -2\hat{i} + 3\hat{j} - 3\hat{k}$, $\vec{b} = 4\hat{i} + 5\hat{j} + (\lambda - 10)\hat{k}$, and $\vec{c} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ is $11$ cubic units, then the sum of the possible values of $\lambda$ is

If $a \cdot b = b \cdot c = c \cdot a = 0$,then what is the value of the scalar triple product $[a b c]$?

The vector $c \cdot (b+c) \times (a+b+c)$ is equal to

If $\overline{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, \overline{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$ and $\overline{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$ are non-zero non-coplanar vectors and $m$ is a non-zero scalar such that $[m\overline{a}+\overline{b} \quad m\overline{b}+\overline{c} \quad m\overline{c}+\overline{a}] = 28[\overline{a} \quad \overline{b} \quad \overline{c}]$,then the value of $m$ is: