Three Dimension...

The angle between the straight lines $$x - 1 = \dfrac{2y + 3}{3} = \dfrac{z + 5}{2}$$ and $$x = 3r + 2;\, y = -2r - 1; \,z = 2$$, where $$r$$ is a parameter, is

If the two lines $$\cfrac { x-1 }{ 2 } =\cfrac { 1-y }{ -a } =\cfrac { z }{ 4 } $$ and $$\cfrac { x-3 }{ 1 } =\cfrac { 2y-3 }{ 4 } =\cfrac { z-2 }{ 2 } $$ are perpendicular, then the value of $$a$$ is equal to

The number of straight lines that are equally inclined to the three-dimensional coordinate axes, is

If the direction ratios of two lines are given by $$3lm-4ln+mn=0$$ and $$l+2m+2n=0$$, then, the angle between the lines is

The equation of the plane perpendicular to the line $$\cfrac { x-1 }{ 1 } =\cfrac { y-2 }{ -1 } =\cfrac { z+1 }{ 2 } $$ and passing through the point $$(2,3,1)$$ is

The angle between two diagonals of a cube will be

If a line makes the angle $$\alpha ,\beta ,\gamma $$ with three dimensional coordinate axes respectively, then $$\cos { 2\alpha  } +\cos { 2\beta  } +\cos { 2\gamma  } $$ is equal to

The point $$P(x, y, z)$$ lies in the first octant and its distance from the origin is $$12$$ units. If the position vector of $$P$$ make $$45^{\circ}$$ and $$60^{\circ}$$ with the x-axis and y-axis respectively, then the coordinates of $$P$$ are

If the direction cosines of a line are $$\left (\dfrac {1}{c}, \dfrac {1}{c}, \dfrac {1}{c}\right )$$, then

A line makes the same angle $$\theta $$ with each of the $$x$$ and $$z$$-axes. If the angle $$\beta$$, which it makes with $$y$$-axis, is such that $$\sin ^{ 2 }{ \beta  } =3\sin ^{ 2 }{ \theta  } $$, then $$\cos ^{ 2 }{ \theta  } $$ is equal to