The AreParallel command determines if two lines or two planes are parallel or not, or if a line and a plane are parallel.

The Line can be either in 2D or 3D.

Two Line objects in 3D can be non-intersecting but not parallel to each other.  This property between such lines, which is called skewness, can be detected using AreSkew.

Two identical lines or two identical planes are parallel to each other. Similarly, a line that lies in a plane is parallel to that plane.

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

$\mathrm{l1}≔\mathrm{Line}\left(\left[0\,2\,6\right]\,⟨10\,15\,20⟩\right)\:$

$\mathrm{l2}≔\mathrm{Line}\left(\left[-3\,7\,3\right]\,⟨2\,3\,4⟩\right)\:$

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

$\mathrm{true}$

$\mathrm{l3}≔\mathrm{Line}\left(\left[3\,2\,3\right]\,⟨-2\,6\,1⟩\right)\:$

$\mathrm{AreParallel}\left(\mathrm{l1}\,\mathrm{l3}\right)$

$\mathrm{false}$

$\mathrm{l4}≔\mathrm{Line}\left(\left[0\,-1\,-2\right]\,⟨1\,2\,3⟩\right)\:$

$\mathrm{l5}≔\mathrm{Line}\left(\left[-3\,-2\,-1\right]\,⟨-4\,-3\,2⟩\right)\:$

$\mathrm{AreParallel}\left(\mathrm{l4}\,\mathrm{l5}\right)$

$\mathrm{false}$

$\mathrm{Intersects}\left(\mathrm{l4}\,\mathrm{l5}\right)$

$\mathrm{false}$

$\mathrm{AreSkew}\left(\mathrm{l4}\,\mathrm{l5}\right)$

$\mathrm{true}$

$\mathrm{l6}≔\mathrm{Line}\left(\left[-1\,-3\right]\,\left[4\,5\right]\right)\:$

$\mathrm{l7}≔\mathrm{Line}\left(\left[0\,8\right]\,⟨5\,8⟩\right)\:$

$\mathrm{AreParallel}\left(\mathrm{l6}\,\mathrm{l7}\right)$

$\mathrm{true}$

$\mathrm{p1}≔\mathrm{Plane}\left(\left[8\,2\,6\right]\,⟨-1\,-6\,4⟩\right)\:$

$\mathrm{p2}≔\mathrm{Plane}\left(\left[-2\,8\,-6\right]\,⟨1\,6\,-4⟩\right)\:$

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

$\mathrm{true}$

$\mathrm{p3}≔\mathrm{Plane}\left(\left[3\,7\,1\right]\,\left[7\,1\,3\right]\,\left[-3\,6\,-10\right]\right)\:$

$\mathrm{AreParallel}\left(\mathrm{p1}\,\mathrm{p3}\right)$

$\mathrm{false}$

The Student[MultivariateCalculus][AreParallel] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.