If $\alpha, \beta, \gamma$ are the direction angles of the line $x = 4z + 3, y = 2 - 3z$, then the value of $\cos \alpha + \cos \beta + \cos \gamma$ is...

The point of intersection of the lines $l_1: r(t) = (i - 6j + 2k) + t(i + 2j + k)$ and $l_2: R(u) = (4j + k) + u(2i + j + 2k)$ is

Show that the line joining the origin to the point $(2,1,1)$ is perpendicular to the line determined by the points $(3,5,-1)$ and $(4,3,-1).$

$A$ line $L_1$ passes through the point with position vector $3 \hat{i}$ and is parallel to the vector $-\hat{i}+\hat{j}+\hat{k}$. Another line $L_2$ passes through the point with position vector $\hat{i}+\hat{j}$ and is parallel to the vector $\hat{i}+\hat{k}$. Find the position vector of the point of intersection of lines $L_1$ and $L_2$.

If $\theta$ is the angle between the lines $\frac{x - 1}{2} = \frac{2y + 3}{4}; z = -2$ and $x = 1; \frac{y - 1}{2} = \frac{z + 1}{2}$, then