angle between line and plane

The angle between a line l and a plane τ is defined as the least possible angle ω between l and a line contained by τ.

It is apparent that ω satisfies always  0ω90.

Let the plane τ be given by the equation (  Ax+By+Cz+D=0,  i.e. its normal vectorMathworldPlanetmath has the componentsPlanetmathPlanetmathPlanetmath A,B,C. Let a direction vector of the line l have the components a,b,c. Then the angle ω between l and τ is obtained from the equation


In fact, the right hand side ( is the cosine of the angle α between l and the surface normal of τ (see angle between two lines), and ω is the complementary angleMathworldPlanetmath of α.


Example.  Consider the xy-plane and the line l through the origin and the point  (1, 1, 1). We can use the components 1, 1, 1 for the direction vector of l and the components 0, 0, 1 for the normal vector of the plane. We have

Title angle between line and plane
Canonical name AngleBetweenLineAndPlane
Date of creation 2013-03-22 17:30:14
Last modified on 2013-03-22 17:30:14
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Definition
Classification msc 51N20
Synonym slant
Synonym inclination
Related topic AngleBetweenTwoLines
Related topic DotProduct
Related topic EquationOfPlane
Related topic AngleBetweenTwoPlanes
Related topic NormalOfPlane
Related topic ProjectionOfRightAngle
Defines angle between plane and line