Vinod Singh and Shweta Pawar Solutions for Chapter: Three Dimensional Geometry, Exercise 3: Competitive Thinking

The points $A\left(1,0,1\right), B\left(-1,2,3\right)$ and $C\left(0,1,2\right)$ are

The angle between pair of lines whose direction ratios are $<1, 1, 2>$  and $<\sqrt{3}-1, -\sqrt{3}-1, 4>$ is 

The angle between a line with direction ratios $2, 2, 1$ and a line joining $\left(3,1,4\right)$ to $\left(7,2,12\right)$ is

If the points $A, B, C, D$ are $\left(2,3,-1\right), \left(3,5,-3\right), \left(1,2,3\right), \left(3,5,7\right)$, respectively, then the angle between $AB$ and $CD$ is

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

The direction cosines of a line, which is perpendicular to both the lines whose direction ratios are $-1,2,2$ and $0,2,1$, are

The direction cosines of the line, which is perpendicular to the lines whose direction cosines are proportional to $<1,-1,2>$ and $<2,1,-1>$, are:

If a line makes $\alpha , \beta , \gamma , \delta$ angles with four diagonals of a cube, then $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +\cos 2\delta =$