Projects - Cengage Learning

C-78 Appendix C
PROJECT
Lines, Circles, and Ray Tracing with Vectors
In the x-y plane, a line and its equations are uniquely determined by the line’s
slope (or undefined slope if the line is vertical) and the coordinates of one point
on the line. We know an equation of the line with slope m and y-intercept b is
y mx b.
Our first goal in this project is to find a vector equation for a line. Let P 0 be
a point in the x-y plane with A a nonzero vector. In Figure A we draw the position
vector P 0 from the origin to the point P 0 and the vector A with initial point
P 0 , and then draw the vectors P 0 A, P 0 2A, P 0 3A, P 0 (1)A, and
P 0 (2)A.
Figure A shows that each point P on the line has a description of the form
P 0 tA for some real number t. As t varies over all real numbers we sweep out
all points on the line. These observations give us a vector equation for the line
y
_A
_A
P0 +(_2)A
P 0 +(_1)A
0
P
P 0
P 0
+A
A
P 0 +2A
A
A
P 0 +3A
P= P 0 +tA
P
x
Figure A
through the point P 0 with direction vector A. (Note: A direction vector for a
line is a nonzero vector that is parallel to the line.) Letting P denote the position
vector from the origin to the arbitrary point P on the line we have
P P 0 tA for q t q (1)
EXAMPLE
SOLUTION
A Vector Equation for a Line
Consider the line in the x-y plane passing through the point (2, 1) with
slope 2.
(a) Sketch the line and find its slope-intercept equation.
(b) Find a vector equation for the line.
(c) Find the points on the line corresponding to t 1, 0, and 2.
(a) The line is sketched in Figure B. It has an equation of the form
y mx b
Here m 2, so y 2x b. (2, 1) lies on the line, so
Therefore y 2x 5.
1 2(2) b
b 5