If $(2, 3, c)$ are the direction ratios of a ray passing through the point $C(5, q, 1)$ and also the midpoint of the line segment joining the points $A(p, -4, 2)$ and $B(3, 2, -4)$,then $c \cdot (p + 7q) = $

Find the direction cosines of the line passing through the points $P(2, 3, 5)$ and $Q(-1, 2, 4)$.

The direction cosines of the line which is perpendicular to the lines with direction cosines proportional to $\langle 1, -2, -2 \rangle$ and $\langle 0, 2, 1 \rangle$ is given by

Suppose $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$ are the direction cosines of two lines and $\theta$ is the angle between them,where $\cos \theta = \pm(l_1 l_2 + m_1 m_2 + n_1 n_2)$. Let $A=(1, -2, 3)$,$B=(3, 1, -3)$,and $C=(-3, 1, 3)$ be the vertices of $\triangle ABC$. Then,$\cos A =$

The direction cosines of the line passing through $P(2, 3, -1)$ and the origin $O(0, 0, 0)$ are:

What are the direction cosines of a line perpendicular to the $yz$-plane?