Which of the following is not equal to $w \cdot(u \times v)$ ?

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 the origin and the points $(1, 2, 3)$,$(2, 3, 4)$,and $(x, y, z)$ are coplanar,then

If the vectors $\vec{a}=\lambda \hat{i}+\mu \hat{j}+4 \hat{k}$,$\vec{b}=2 \hat{i}+4 \hat{j}-2 \hat{k}$ and $\vec{c}=2 \hat{i}+3 \hat{j}+\hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units,then the sum of all possible values of $\lambda+\mu$ is equal to:

If the points whose position vectors are $3i - 2j - k,$ $2i + 3j - 4k,$ $-i + j + 2k,$ and $4i + 5j + \lambda k$ lie on a plane,then $\lambda = $

If $\overline{a}$ and $\overline{c}$ are unit vectors inclined at $\frac{\pi}{3}$ with each other and $(\overline{a} \times (\overline{b} \times \overline{c})) \cdot (\overline{a} \times \overline{c}) = 5$,then the value of $5[\overline{a} \overline{b} \overline{c}] = $