PlanetMath: angle between line and plane

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angle between line and plane

(Definition)

The angle between a line and a plane $\tau$ is defined as the least possible angle $\omega$ between 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 $Ax\!+\!By\!+\!Cz\!+\!D = 0$ , i.e. its normal vector has the components $A,\,B,\,C$ . Let a direction vector of the line have the components $a,\,b,\,c$ . Then the angle $\omega$ between and $\tau$ is obtained from the equation

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

In fact, the right hand side is the cosine of the angle $\alpha$ between

and the surface normal of $\tau$ (see angle between two lines), and $\omega$ is the complementary angle of $\alpha$ .

$\begin{pspicture}(-1,-0.5)(8,5) \psline(2,3)(0,0)(5,0)(7,3) \psline(2,3)(3.44,3)... ...0,-0.032)(5.023,-0.032)(7.023,3) \psarc(4.7,1.5){0.25}{130}{180} \end{pspicture}$

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

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

"angle between line and plane" is owned by pahio.

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Other names:

slant, inclination

Also defines:

angle between plane and line

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Cross-references: point, origin, complementary angle, angle between two lines, surface normal, cosine, equation, direction vector, components, normal vector, contained, plane, line, angle
There is 1 reference to this entry.

This is version 9 of angle between line and plane, born on 2007-08-25, modified 2008-07-27.
Object id is 9893, canonical name is AngleBetweenLineAndPlane.
Accessed 3260 times total.

Classification:

AMS MSC:

51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)

Pending Errata and Addenda

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