Find the direction cosines of the vector joining the points $A(1, 2, -3)$ and $B(-1, -2, 1)$,directed from $A$ to $B$.

The direction cosines of two lines satisfy the equations $l+m+n=0$ and $2mn+3ln-5lm=0$. The angle between these lines is

$A$ line makes angles $\frac{\alpha}{2}, \frac{\beta}{2}, \frac{\gamma}{2}$ with the positive directions of the coordinate axes ($x, y, z$ axes respectively). Then,the value of $\cos \alpha + \cos \beta + \cos \gamma$ is:

Find the angle between the lines whose direction ratios are $1, 1, 2$ and $\sqrt{3}-1, -\sqrt{3}-1, 4$. (in $^{\circ}$)

If the direction cosines of a line $L$ are $(pq, q, q)$ and the angle between the line $L$ and the positive direction of the $X$-axis is $\frac{\pi}{3}$,then $p^2 : q^2 =$

The angle between the lines whose direction ratios satisfy the equations $l+m+n=0$ and $l^2=m^2+n^2$ is