CS 418: Homework #1
Assigned: Friday February 4, 2005
Due: Tuesday February 15, 2005
This homework is meant to exercise some basic math skills that will come in handy later. Much of the linear
algebra and geometric knowledge you need can be found in the textbook. You should return your completed
assignment at the end of class on Tuesday, February 15.
Please be organized when writing your answers to these questions. Make sure that all solutions are
clearly indicated and labelled with the question they are answering. Remember to write clearly and legibly.
Unreadable answers will receive 0 credit.
For the purposes of this homework, and indeed the rest of the course, you should assume that points/directions
are represented by column vectors and that we are using a right-handed coordinate system.
1. Let p1,p2,p3be three points in 3-D Euclidean space. We can linearly interpolate between pairs of these
points as follows:
p12(t) = (1 −t)p1+tp20≤t≤1
p23(t) = (1 −t)p2+tp3
Now suppose that we want to construct:
p123(t) = (1 −t)p12 (t) + tp23(t)
Give an expression for p123 as a linear combination of the points p1,p2,p3and t.
2. Suppose that you are given two unit vectors: an up direction uand a forward direction f.
(a) Give an expression for the backward direction, involving only uand/or f.
(b) Give an expression for the left direction, involving only uand/or f.
(c) Suppose that you are standing at the point p. You wish to move forward to a new point p0which
is dunits from p. Give an expression for p0involving only d,u, and/or f.
3. Given a point qand direction vector v, we can define the line through qin the direction vby the
parametric equation:
p(t) = q+tv
In other words, the line is the set of all points p(t) for any real value of t.
Suppose that you are given a point rthat is not on the line p(t). Derive an equation for the perpendicular
distance of rto the line.
