Three-dimension...

is a vector joining two points P(x₁ , y₁ , z₁ ) and Q(x₂ , y₂ , z₂ ). If  Direction cosines of  are

Shortest distance between the lines  

Find the shortest distance between the lines    and  

The distance d from a point P(x₁ , y₁ , z₁ ) to the plane Ax + By + Cz + D = 0 is

Determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.7x + 5y + 6z + 30 = 0 and 3x –y –10z + 4 = 0

If l, m and n are the direction cosines of a line, Direction ratios of the line are the numbers which are

Distance between  

Find the angle between the following pairs of lines:   and  

Determine the direction cosines of the normal to the plane and the distance from the origin. Plane z = 2

In the following case, determine whether the given planes are parallel orperpendicular, and in case they are neither, find the angles between them. 2x + y + 3z –2 = 0 and x –2y + 5 = 0

If l, m, n are the direction cosines and a, b, c are the direction ratios of a line then

If a line makes angles  90^∘ , 135^∘ , 45^∘ with the x, y and z –axes respectively, find its direction cosines.

In the vector form, equation of a plane which is at a distance d from the origin, and   is the unit vector normal to the plane through the origin is

Determine the direction cosines of the normal to the plane and the distance from the origin. Plane x + y + z = 1

In the following case, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. 2x –2y + 4z + 5 = 0 and 3x –3y + 6z –1 = 0