The parallelopiped is determined by vectors $\bar{a} = -2\hat{i} + 5\hat{j} + 3\hat{k}$, $\bar{b} = \hat{i} + 3\hat{j} - 2\hat{k}$, $\bar{c} = -3\hat{i} + \hat{j} + 4\hat{k}$. The altitude of parallelopiped on the parallelogram base determined by vectors $\bar{b}$ and $\bar{c}$ is

The vector equation of the plane passing through the point $A(-2, 7, 5)$ and parallel to the vectors $4\hat{i} - \hat{j} + 3\hat{k}$ and $\hat{i} + \hat{j} + \hat{k}$ is ...

The point of intersection of the two lines $\dfrac{x-3}{3} = \dfrac{y-3}{-1}, z - 1 = 0$ and $\dfrac{x-6}{2} = \dfrac{z-1}{3}, y - 2 = 0$ is....

If the centroid of a tetrahedron $OABC$ is $(1, 2, -1)$, where O is the origin, $A(a, 2, 3), B(1, b, 2), C(2, 1, c)$ are the other vertices, then the distance of the point $P(a, b, c)$ from the origin is...