If the line passing through the points $(a, 1, 6)$ and $(3, 4, b)$ crosses the $yz$-plane at the point $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$,then:

If two lines $L_1$ and $L_2$ in space are defined by $L_1 = \{ x = \sqrt{\lambda} y + (\sqrt{\lambda} - 1), z = (\sqrt{\lambda} - 1)y + \sqrt{\lambda} \}$ and $L_2 = \{ x = \sqrt{\mu} y + (1 - \sqrt{\mu}), z = (1 - \sqrt{\mu})y + \sqrt{\mu} \}$,then $L_1$ is perpendicular to $L_2$ for all non-negative reals $\lambda$ and $\mu$ such that:

Find the coordinates of the point where the line passing through $(5, 1, 6)$ and $(3, 4, 1)$ crosses the $ZX$-plane.

The angle between the lines $\frac{x}{1} = \frac{y}{0} = \frac{z}{-1}$ and $\frac{x}{3} = \frac{y}{4} = \frac{z}{5}$ is

Let the point,on the line passing through the points $P(1, -2, 3)$ and $Q(5, -4, 7)$,farther from the origin and at a distance of $9$ units from the point $P$,be $(\alpha, \beta, \gamma)$. Then $\alpha^2 + \beta^2 + \gamma^2$ is equal to:

The angle between the pair of lines $\vec{r} = -3\hat{i} + \hat{j} + 3\hat{k} + \lambda(3\hat{i} + 5\hat{j} + 4\hat{k})$ and $\vec{r} = -\hat{i} + 4\hat{j} + 5\hat{k} + \mu(\hat{i} + \hat{j} + 2\hat{k})$ is . . . . . . .