line - Maple Help

geom3d

 line
 define a line

 Calling Sequence line(l, [A, B] ) line(l, [A, v] ) line(l, [A, dseg] ) line(l, [A, p1] ) line(l, [p1, p2] ) line(l, [$\mathrm{b1}t+\mathrm{a1}$, $\mathrm{b2}t+\mathrm{a2}$, $\mathrm{b3}t+\mathrm{a3}$ ], t)

Parameters

 l - the name of the line A, B - points v - vector dseg - directed line segment p1, p2 - planes a1, a2, a3, b1, b2, b3 - algebraic expressions t - (optional) a name denoting the parameter in the parametric equations of the line

Description

 • A line l can be defined as follows:
 + from two given points A and B
 + from a given point A and a vector v of dimension 3 or a directed segment seg. The line defined is the line that passes through A and has v as its direction-ratios.
 + from a given point A and a plane p1. The line defined is the line that passes through A and perpendicular to the plane p1.
 + from two given planes p1 and p2. The line defined is the line of intersection of two planes p1 and p2 (if exists).
 + from the parametric equations of the line $\left[\mathrm{b1}t+\mathrm{a1},\mathrm{b2}t+\mathrm{a2},\mathrm{b3}t+\mathrm{a3}\right]$. If the third optional argument t is not given, and if a name is assigned to the environment variable _EnvTName, then this name will be used as the name of the parameter in the parametric equations of the line. Otherwise, Maple will prompt the user to input the name of the parameter.
 • To access the information relating to a line l, use the following function calls:

 form(l) returns the form of the geometric object (i.e., line3d if l is a line). FixedPoint(l) returns a fixed point on l. ParallelVector(l) returns a direction-ratios of l. DirectionRatios(l) returns the direction-ratios of l. Equation(l) returns the parametric equations that represents the line l. xname(l), yname(l), or zname(l) returns the name of the x-axis; y-axis, z-axis or FAIL if the axis is not assigned to any name. detail(l) returns a detailed description of the line l.

 • The command with(geom3d,line) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{geom3d}\right):$

Find the equation of the line through [1,2,-1] perpendicular to the plane $3x-5y+4z=5$,the length of the perpendicular, and the coordinates of its foot

 > $\mathrm{point}\left(A,1,2,-1\right),\mathrm{plane}\left(p,3x-5y+4z=5,\left[x,y,z\right]\right):$

The parametric equations of the line l with parameter t is

 > $\mathrm{line}\left(l,\left[A,p\right]\right)$
 ${l}$ (1)
 > $\mathrm{Equation}\left(l,'t'\right)$
 $\left[{1}{+}{3}{}{t}{,}{2}{-}{5}{}{t}{,}{-}{1}{+}{4}{}{t}\right]$ (2)

The projection of $A$ on the plane is

 > $\mathrm{projection}\left(B,A,p\right)$
 ${B}$ (3)
 > $\mathrm{coordinates}\left(B\right)$
 $\left[\frac{{49}}{{25}}{,}\frac{{2}}{{5}}{,}\frac{{7}}{{25}}\right]$ (4)

The length of the perpendicular is

 > $\mathrm{distance}\left(A,B\right)$
 $\frac{{8}{}\sqrt{{2}}}{{5}}$ (5)

Let the straight line pass through the point $A=\left[\mathrm{x1},\mathrm{y1},\mathrm{z1}\right]$ and has direction-cosines (or ratios) $\left[l,m,n\right]$

 > $\mathrm{assume}\left(l\ne 0\right):$
 > $\mathrm{point}\left(A,\left[\mathrm{x1},\mathrm{y1},\mathrm{z1}\right]\right):$$v≔\left[l,m,n\right]:$

Define the line l1 that passes through $A$ and has $\left[l,m,n\right]$ as its direction-ratios

 > $\mathrm{line}\left(\mathrm{l1},\left[A,v\right]\right):$
 > $\mathrm{detail}\left(\mathrm{l1}\right)$
 $\begin{array}{ll}{\text{name of the object}}& {\mathrm{l1}}\\ {\text{form of the object}}& {\mathrm{line3d}}\\ {\text{equation of the line}}& \left[{\mathrm{_x}}{=}{\mathrm{_t}}{}{\mathrm{l~}}{+}{\mathrm{x1}}{,}{\mathrm{_y}}{=}{\mathrm{_t}}{}{m}{+}{\mathrm{y1}}{,}{\mathrm{_z}}{=}{\mathrm{_t}}{}{n}{+}{\mathrm{z1}}\right]\end{array}$ (6)
 > $l≔'l':$

Define the line l which is the intersection of two given planes p1, p2

 > $\mathrm{plane}\left(\mathrm{p1},4x+4y-5z=12,\left[x,y,z\right]\right):$
 > $\mathrm{plane}\left(\mathrm{p2},8x+12y-13z=32,\left[x,y,z\right]\right):$
 > $\mathrm{line}\left(l,\left[\mathrm{p1},\mathrm{p2}\right]\right)$
 ${l}$ (7)
 > $\mathrm{Equation}\left(l,'t'\right)$
 $\left[{1}{+}{8}{}{t}{,}{2}{+}{12}{}{t}{,}{16}{}{t}\right]$ (8)