# 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  $(1,\,1,\,1)$. 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

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