angle between line and plane

The angle between a line $l$ and a plane $\tau$ is defined as the least possible angle $\omega$ between $l$ and a line contained by $\tau$.

It is apparent that $\omega$ satisfies always  $0\leqq \omega \leqq {90}^{\circ }$.

Let the plane $\tau$ be given by the equation (http://planetmath.org/EquationOfPlane)  $Ax+By+Cz+D=0$,  i.e. its normal vector has the components $A,B,C$. Let a direction vector of the line $l$ have the components $a,b,c$. Then the angle $\omega$ between $l$ and $\tau$ is obtained from the equation

$$\sin \omega =\frac{|Aa+Bb+Cc|}{\sqrt{A^2+B^2+C^2}\sqrt{a^2+b^2+c^2}}.$$

In fact, the right hand side (http://planetmath.org/Equation) is the cosine of the angle $\alpha$ between $l$ and the surface normal of $\tau$ (see angle between two lines), and $\omega$ is the complementary angle of $\alpha$.

Example.  Consider the $xy$-plane and the line $l$ through the origin and the point  $(1,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}1})$. We can use the components $1,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}1}$ for the direction vector of $l$ and the components $0,\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}1}$ for the normal vector of the plane. We have

$$\omega =\arcsin \frac{1\cdot 0+1\cdot 0+1\cdot 1}{\sqrt{1^2+1^2+1^2}\sqrt{0^2+0^2+1^2}}=\arcsin \frac{1}{\sqrt{3}}\approx {35.26}^{\circ }.$$

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