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.

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 $\left[0\,\mathrm{\pi }\right]$.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{MultivariateCalculus}\right]\right)\:$

$\mathrm{v1}≔⟨1\,2\,3⟩\:$

$\mathrm{v2}≔⟨0\,0\,1⟩\:$

$\mathrm{v3}≔⟨a\,b\,c⟩\:$

$\mathrm{l1}≔\mathrm{Line}\left(\left[0\,0\,0\right]\,⟨1\,2\,4⟩\right)\:$

$\mathrm{l2}≔\mathrm{Line}\left(\left[1\,1\,2\right]\,⟨2\,3\,0⟩\right)\:$

$\mathrm{p1}≔\mathrm{Plane}\left(\left[1\,2\,0\right]\,⟨-1\,1\,0⟩\right)\:$

$\mathrm{p2}≔\mathrm{Plane}\left(\left[1\,1\,2\right]\,⟨1\,2\,1⟩\right)\:$

$\mathrm{Angle}\left(\mathrm{v1}\,\mathrm{v2}\right)$

$\arccos \left(\frac{3\sqrt{14}}{14}\right)$

$\mathrm{Angle}\left(\mathrm{v3}\,\mathrm{l1}\right)$

$\min \left(\mathrm{\pi }-\arccos \left(\frac{\left(a+2b+4c\right)\sqrt{21}}{21\sqrt{a^2+b^2+c^2}}\right)\,\arccos \left(\frac{\left(a+2b+4c\right)\sqrt{21}}{21\sqrt{a^2+b^2+c^2}}\right)\right)$

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

$0$

$\mathrm{Angle}\left(\mathrm{l1}\,\mathrm{l2}\right)$

$\arccos \left(\frac{8\sqrt{21}\sqrt{13}}{273}\right)$

$\mathrm{Angle}\left(\mathrm{l2}\,\mathrm{p1}\right)$

$\frac{\mathrm{\pi }}{2}-\arccos \left(\frac{\sqrt{13}\sqrt{2}}{26}\right)$

$\mathrm{Angle}\left(\mathrm{p1}\,\mathrm{p2}\right)$

$\arccos \left(\frac{\sqrt{2}\sqrt{6}}{12}\right)$

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

For more information on Maple 18 changes, see Updates in Maple 18.