Angle between a line and a plane

Angle between a line and a plane

We know that the angle between a line and a plane is the  complement of the angle between the normal to the plane and the line

Let  =  + t be the equation of the line and â‹…  = p be the equation of the plane. We know that  is parallel to the given line and  is normal to the given plane. If θ is the acute angle between the line and the plane, then the acute angle between  and  is ((Ï€/2)-θ).Therefore,

So, the acute angle between the line and the plane is given by θ =           ….(1)

In Cartesian form if  and ax + by + cz = p are the equations of the line and the plane, then  = a₁iˆ + b₁ ˆj + c₁kˆ and   = aˆi + bˆj + cˆk . Therefore, using (1), the acute angle θ between the line and plane is given by

Remark

(i) If the line is perpendicular to the plane, then the line is parallel to the normal to the plane.

So,  is perpendicular to . Then we have  = Î».  Î» ∈ R ,which gives 

(ii) If the line is parallel to the plane, then the line is perpendicular to the normal to the plane.

Therefore, . = 0 ⇒ aa₁ + bb₁ + cc₁ = 0

Example 6.48

Find the angle between the straight line  = (2ˆi + 3ˆj + ˆk )+ t (ˆi - ˆj + ˆk ) and the plane 2x - y + z = 5 .

Solution

The angle between a line  =  + t  and a plane â‹…  = p with normal  is Î¸