A Self-guided Tour of Parametric Equations and Vectors
1.
The parametric equations x = −2−3t and y = 6+4t describe the position of a particle, in meters and
seconds. How does the particle’s position change each second? each minute? What is the speed of
the particle, in meters per second? Write parametric equations that describe this motion, using
meters and minutes as units.
2.
Given the vector [−5, 12], find the following vectors:
a) same direction, twice as long
b) same direction, length 1
c) opposite direction, length 10
3.
Motions of three particles are described by the following three pairs of equations
a) x = 2− 2t
y = 5+7t
b) x = 4− 2t
y = −2 +7t
c) x = 2− 2(t + 1)
y = 5+7(t + 1)
Describe the motion of each particle based on the parametric equations. How do the positions of
these particles compare at any given moment?
4.
Robin is moving on the coordinate plane according to the rule (x, y) = (−3+8t, 5+6t), where
distance is measured in km and time is measured in hours. Casey is following 20 km behind, at the
same speed. Write parametric equations describing Casey’s motion.
5.
Graph the line that is described parametrically by (x, y) = (2t, 4 − t), then
a) confirm that the point corresponding to t = 0 is exactly 5 units from (3, 8);
b) write a formula in terms of t for the distance from (3, 8) to (2t, 4 − t);
c) find the other point on the line that is 5 units from (3, 8);
d) find the point on the line that minimizes the distance to (3, 8).
6.
Show that two vectors a, b and c, d are perpendicular if, and only if, ac + bd = 0. The number
ac + bd is called the dot product of the vectors a, b and c, d .
Hint: draw the vectors in standard position and use the Pythagorean theorem. The number ac + bd
is called the dot product of the vectors a, b and c, d .
7.
Given points A = (0, 0) and B = (−2, 7), find coordinates for C and D so that ABCD is a square.
A Self-guided Tour of Parametric Equations and Vectors
8.
Find k so that the vectors 4, 3 and k , 6 (a) point in the same direction; (b) are perpendicular.
9.
Translate the line 5x + 7y = 35 by vector 3,10 . Find an equation for the new line.
10. Find the point of intersection of the lines Pt = (−1+3t,3+2t) and Qr = (4−r,1+2r).more here
11. Draw a parallelogram whose adjacent edges are determined by vectors 2,5 and 7, 1 , placed
so that they have a common initial point. This is called placing vectors tail-to-tail. Find the area of
the parallelogram.
12. To the nearest tenth of a degree, find the angle formed by placing the vectors 4,3 and 7,1
tail to tail.
13. What graph is traced by the parametric equation ( x, y )  (t , 4  t 2 ) ?
14. Find components for the vector that points from (1, 1, 1) to (2, 3, 4). Then find the distance from
(1, 1, 1) to (2, 3, 4) by finding the length of this vector.
15. Show that the vectors 4, 5, 3 and 2, 1,1 are perpendicular.
16. Find components for a vector of length 21 that points in the same direction as 2,3, 6 .
17. Find coordinates for the point on segment KL that is 5 units from K, where:
a) K = (0, 0, 0) and L = (4, 7, 4);
b) K = (3, 2, 1) and L = (7, 9, 5).
18. Give an example of a nonzero vector that is perpendicular to 5, 7, 4 .
19. Give an example of a vector perpendicular to 6, 2,3 that has the same length.
A Self-guided Tour of Parametric Equations and Vectors
20. An Unidentified Flying Object (UFO) moving along a line with constant speed was sighted at
(8, 9, 10) at noon and at (13, 19, 20) at 1:00 pm. Where was the UFO at 12:20pm? When, and
from where, did it leave the ground (z = 0)? What was the UFO’s speed?
21. Let A = (7, 1, 1) and B = (−3, 2, 7). Find all the points P on the z-axis that make angle APB a
right angle.
22. The position of an airplane that is approaching its airport is described parametrically by:
Pt  (1000,500,900)  t 100, 50, 90 . For what value of t is the airplane closest to the traffic
control center located at (34,68,16)?
23. Use a vector to determine whether the points (2, 5, 7), (12, 25, 37), and (27, 55, 81) collinear.
24. Think of the ground you are standing on as the xy-plane. The vector 12, 5,10 points from you
toward the Sun. How high is the Sun in the sky?
25. Do the lines (x, y, z) = (5+2t,3 + 2t, 1 − t) and (x, y, z) = (13 − 3r, 13 − 4r, 4 − 2r) intersect? If so,
at what point? If not, how do you know?
26. Let A = (1, 2, 3), B = (3, 7, 9), and D = (−2, 3,−1). Find coordinates for vertex C of parallelogram
ABCD. How many parallelograms can you find that have the three given vertices among their four
vertices?
27. Brett and Jordan are cruising, according to the equations Bt = (27+4t, 68−7t, 70+4t) and
Jt = (23+4t, 11 + t, 34 + 8t). Show that their paths intersect, but that there is no collision. Who
reaches the intersection first? Who is moving faster?
28. Find a vector of length 3 that is perpendicular to 2,1, 2 .
29. Show the lines (x, y, z) = (5+2t, 3+2t, 1− t) and (x, y, z) = (13− 3r, 13− 4r, 4+2r) are not parallel,
and that they do not intersect. Such lines are called skew.
30. The position of a starship is given by the equation Pt = (18+3t, 24+4t, 110−5t). For what values of
t is the starship within 100 units of a space station placed at the origin?