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Student[MultivariateCalculus]

 Angle
 Determine the angle between lines, vectors, and planes.

 Calling Sequence Angle(x, y)

Parameters

 x - a vector, a line, or a plane y - a vector, a line, or a plane

Description

 • The Angle command determines the angle between two vectors, a vector and a line, a vector and a plane, two lines, a line and a plane, or two planes.

Notes

 • The angle between two intersecting lines can be measured at the intersection point; the angle returned is in the interval $\left[0,\frac{\mathrm{\pi }}{2}\right]$. When two lines do not intersect, we define the angle determined by them as the angle between two lines through the origin parallel to the given lines.
 • The angle between two planes is equal to the angle between their normals.
 • The angle between a line and a plane is equal to the complement of the angle between the line and the normal of the plane.
 • An angle involving one vector, v, is the same as if instead of the vector, you had supplied a line having v as its direction. An angle between two vectors is slightly different, in that it can attain all values in $\left[0,\mathrm{\pi }\right]$.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{MultivariateCalculus}]\right):$
 > $\mathrm{v1}≔⟨1,2,3⟩:$
 > $\mathrm{v2}≔⟨0,0,1⟩:$
 > $\mathrm{v3}≔⟨a,b,c⟩:$
 > $\mathrm{l1}≔\mathrm{Line}\left(\left[0,0,0\right],⟨1,2,4⟩\right):$
 > $\mathrm{l2}≔\mathrm{Line}\left(\left[1,1,2\right],⟨2,3,0⟩\right):$
 > $\mathrm{p1}≔\mathrm{Plane}\left(\left[1,2,0\right],⟨-1,1,0⟩\right):$
 > $\mathrm{p2}≔\mathrm{Plane}\left(\left[1,1,2\right],⟨1,2,1⟩\right):$

Angle between two vectors

 > $\mathrm{Angle}\left(\mathrm{v1},\mathrm{v2}\right)$
 ${\mathrm{arccos}}{}\left(\frac{{3}}{{14}}{}\sqrt{{14}}\right)$ (1)

Angle between a vector and a line

 > $\mathrm{Angle}\left(\mathrm{v3},\mathrm{l1}\right)$
 ${\mathrm{min}}{}\left({\mathrm{π}}{-}{\mathrm{arccos}}{}\left(\frac{{1}}{{21}}{}\frac{\left({a}{+}{2}{}{b}{+}{4}{}{c}\right){}\sqrt{{21}}}{\sqrt{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}}\right){,}{\mathrm{arccos}}{}\left(\frac{{1}}{{21}}{}\frac{\left({a}{+}{2}{}{b}{+}{4}{}{c}\right){}\sqrt{{21}}}{\sqrt{{{a}}^{{2}}{+}{{b}}^{{2}}{+}{{c}}^{{2}}}}\right)\right)$ (2)

Angle between a vector and a plane

 > $\mathrm{Angle}\left(\mathrm{v2},\mathrm{p1}\right)$
 ${0}$ (3)

Angle between two lines

 > $\mathrm{Angle}\left(\mathrm{l1},\mathrm{l2}\right)$
 ${\mathrm{arccos}}{}\left(\frac{{8}}{{273}}{}\sqrt{{21}}{}\sqrt{{13}}\right)$ (4)

Angle between a line and a plane

 > $\mathrm{Angle}\left(\mathrm{l2},\mathrm{p1}\right)$
 $\frac{{1}}{{2}}{}{\mathrm{π}}{-}{\mathrm{arccos}}{}\left(\frac{{1}}{{26}}{}\sqrt{{13}}{}\sqrt{{2}}\right)$ (5)

Angle between two planes

 > $\mathrm{Angle}\left(\mathrm{p1},\mathrm{p2}\right)$
 ${\mathrm{arccos}}{}\left(\frac{{1}}{{12}}{}\sqrt{{2}}{}\sqrt{{6}}\right)$ (6)

Compatibility

 • The Student[MultivariateCalculus][Angle] command was introduced in Maple 18.