Equation of line of intersection of two planes

Equation of line of intersection of two planes

Let  â‹… = p and  â‹… = q  be two non-parallel planes. We know that  and  are perpendicular to the given planes respectively.

So, the line of  intersection of these planes is perpendicular to both  Ã— .    and  . Therefore, it is parallel to the vector  Ã— . Let

Consider the equations of two planes a₁x + b₁y + c₁z = p and a₂x + b₂y + c₂z = q . The line of intersection of the two given planes intersects atleast one of the coordinate planes. For simplicity, we assume that the line meets the coordinate plane z = 0 . Substitute z=0 and obtain the two equations  a₁x + b₁y − p = 0 and a₂ x + b₂y − q = 0 .Then by solving these equations, we get the values of x and y as x₁  and y₁  respectively.

So,  ( x₁ , y₁ , 0) is a point on the required line, which is parallel to  l₁iˆ + l₂ ˆj + l₃kˆ . So, the equation of the line is