# Lines and Planes in 3D

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

> with(linalg):
> 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,

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>