Test: 3-D Geometry (16 Dec) - JEE MCQ

Direction cosines of a line are the cosines of the angles made by the line with the positive direction of the coordinate axis.i.e. x- axis , y-axis and z – axis respectively.

Shortest distance between two skew lines is The line segment perpendicular to both the lines .

On comparing the given equations with :
In the cartesian form two lines


we get ;

x₁ = -1, y₁ = -1,z₁ = -1, ; a₁ = 7, b₁ = -6, c₁ = 1 and 

x₂ = 3, y₂ = 5, z₂ = 7; a₂ = 1, b₂ = -2, c₂ = 1

Now the shortest distance between the lines is given by :

By definition , The angle θ between the planes A₁x + B₁y + C₁z + D₁ = 0 and A₂ x + B₂ y + C₂ z + D₂ = 0 is given by :

The equation of the plane through the line of intersection of the planes

If l, m , n are the direction cosines of a line then , we know that,  l²+ m²+ n² = 1.

On comparing the given equations with: 
, we get: 

The distance of a point whose position vector is  from the plane  given by :

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

In Cartesian coordinate system Shortest distance between the lines

Find the shortest distance between the lines 

On comparing them with :

we get : 

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