Which of the following is not true?

If $\bar{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, \bar{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$,and $\bar{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$,and $[3 \bar{a}+\bar{b} \quad 3 \bar{b}+\bar{c} \quad 3 \bar{c}+\bar{a}] = \lambda \begin{vmatrix} \bar{a} \cdot \hat{i} & \bar{a} \cdot \hat{j} & \bar{a} \cdot \hat{k} \\ \bar{b} \cdot \hat{i} & \bar{b} \cdot \hat{j} & \bar{b} \cdot \hat{k} \\ \bar{c} \cdot \hat{i} & \bar{c} \cdot \hat{j} & \bar{c} \cdot \hat{k} \end{vmatrix}$,then the value of $\lambda$ is:

Let $\vec{a} = \hat{i} - \hat{j}$,$\vec{b} = \hat{j} - \hat{k}$,and $\vec{c} = \hat{k} - \hat{i}$. If $\vec{d}$ is a unit vector such that $\vec{a} \cdot \vec{d} = 0 = [\vec{b} \, \vec{c} \, \vec{d}]$,then find $\vec{d}$.

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

If the vector $\overline{c}$ lies in the plane of $\overline{a}$ and $\overline{b}$,where $\overline{a}=\hat{i}-\hat{j}+2\hat{k}$,$\overline{b}=\hat{i}+\hat{j}+\hat{k}$ and $\overline{c}=x\hat{i}-(2-x)\hat{j}-\hat{k}$,then the value of $x$ is

If $[\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}] = \lambda [\vec{a}, \vec{b}, \vec{c}]^2$,then $\lambda$ is equal to: