. . 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 r tb be the equation of the line and r n be the equation of the plane. We know that b is parallel to the given line and n is normal to the given plane.
If θ is the acute angle between the line and the plane, then the acute angle between n and b is . Therefore, b n b n ° − θ r n tb Fig. . Vector - - So, the acute angle between the line and the plane is given by θ = sin b n b n ...
( ) In Cartesian form if and ax by cz are the equations of the line and the plane, then ˆ a i b j c k ai bj ck . Therefore, using ( ), the acute angle θ between the line and plane is given by aa bb cc − Remark (i) If the line is perpendicular to the plane, then the line is parallel to the normal to the plane. So, b is perpendicular to n . Then we have b where λ ∈ ,which gives (ii) If the line is parallel to the plane, then the line is perpendicular to the normal to the plane.
Therefore, b n aa bb cc ⇒ Example . Find the angle between the straight line t i and the plane The angle between a line r tb and a plane r n with normal n is b n b n − Here, So,we get b n b n −