# Lines and Planes in 3D

```    |\^/|     Maple V Release 3 (SUNY at Albany)
._|\|   |/|_. Copyright (c) 1981-1994 by Waterloo Maple Software and the
\  MAPLE  /  University of Waterloo. All rights reserved. Maple and Maple V
<____ ____>  are registered trademarks of Waterloo Maple Software.
|       Type ? for help.

```

 A line is determined by a direction vector v and a position vector u.

> with(linalg):

```
Warning: new definition for   norm
Warning: new definition for   trace

```

> v := vector(3); u := vector(3);
```
v := array(1 .. 3, [])

u := array(1 .. 3, [])

```

 Yes, this is the way in which maple tells that v and u are two arbitrary vectors in 3 dimensional space. The equation of the line with direction v and passing through u is:

> Line := { u + t*v};

```
Line := {u + t v}

```

 This is the set of all the vectors of the form: "position vector + a scalar multiple of the direction vector". The scalar "t" is a parameter that can take any value from -infinity to infinity. The position vector of every point on the line can be written as a function of the parameter "t" as follows:

> L := t -> u + t*v;

```
L := t -> u + t v

```

> L(0);
```
u

```

> L(1);
```
u + v

```

 To see the coordinates of L(t) in the standard basis, just evalm:

> evalm(L(t));

```
[ u[1] + t v[1], u[2] + t v[2], u[3] + t v[3] ]

```

 The vector L(t) can be interpreted as the position at time "t" of a particle moving with constant velocity vector v that is found at position u at an innitial time 0. The following two commands will display a picture of the line with velocity vector [1,-2,,3] and position vector [2,1,-1].

> with(plots);

```
[animate, animate3d, conformal, contourplot, cylinderplot, densityplot,

display, display3d, fieldplot, fieldplot3d, gradplot, gradplot3d,

implicitplot, implicitplot3d, loglogplot, logplot, matrixplot, odeplot,

pointplot, polarplot, polygonplot, polygonplot3d, polyhedraplot, replot,

setoptions, setoptions3d, spacecurve, sparsematrixplot, sphereplot,

surfdata, textplot, textplot3d, tubeplot]

```

 Notice that the next command "spacecurve" was loaded with the "with" above.

> spacecurve([2+t, 1-2*t, -1+3*t], t=-4..4, axes=NORMAL, color=RED);

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Thu Sep 12 13:02:43 EDT 1996