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

Let $a, b, c$ be distinct non-negative numbers. If the vectors $a\hat{i} + a\hat{j} + c\hat{k}$,$\hat{i} + \hat{k}$,and $c\hat{i} + c\hat{j} + b\hat{k}$ lie in a plane,then $c$ is

For any non-zero vectors $\bar{a}, \bar{b}, \bar{c}$,the value of $\bar{a} \cdot [(\bar{b} \times \bar{c}) \times (\bar{a} + \bar{b} + \bar{c})]$ is

What will be the volume of the parallelepiped whose coterminous edges are given by the vectors $a = i - j + k$,$b = i - 3j + 4k$,and $c = 2i - 5j + 3k$?

If the points having the position vectors $-\hat{i}+4 \hat{j}-4 \hat{k}$,$3 \hat{i}+2 \hat{j}-5 \hat{k}$,$-3 \hat{i}+8 \hat{j}-5 \hat{k}$ and $-3 \hat{i}+2 \hat{j}+\lambda \hat{k}$ are coplanar,then $\lambda=$

The altitude of the parallelepiped,whose coterminus edges are the vectors $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}=2\hat{i}+4\hat{j}-\hat{k}$,and $\bar{c}=\hat{i}+\hat{j}+3\hat{k}$,where $\bar{a}$ and $\bar{b}$ are the sides of the base of the parallelepiped,is: