Math 225: Calculus III	Quiz 9/9/91

Definitions

Needs["Calculus`Advanced`"]

Standard unit vectors:

i = {1, 0, 0};
j = {0, 1, 0};
k = {0, 0, 1};

Projection formula:

A good set of options for 3D graphics output is the following:

SetOptions[Graphics3D,
    Axes->Automatic, BoxStyle->{GrayLevel[0.7]}];

1. Find the parametric equations of the line through the points

p = {-1,2,0};
 q = {4,1,-3};

Solution

The direction of the line is

r = q - p;

The vector equation of the line is:

p + t r

The parametric equations are:

x == -1 + 5t;
y == 2 - t;
z == -3t;

2. Find the equation of the plane through the points

p0 = {2,1,1};
 p1 = {1,2,1};
 p2 = {1,1,2};

Solution

Two vectors parallel to the plane are:

v1 = p1 - p0

v2 = p2 - p0

Their cross product is perpendicular to the plane an thus provide the normal vector N:

n = Cross[v1,v2]

The equation of the plane is:

n.{x,y,z} == n.p0