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Drawing vectors as arrows
A vector in two dimensions can be easily visualized by drawing an arrow
whose length represents its magnitude and whose direction represents its
direction. The x component of a vector can then be visualized as the length
of the shadow it would cast in a beam of light projected onto the x axis, and
similarly for the y component. Shadows with arrowheads pointing back
against the direction of the positive axis correspond to negative compo-
nents.
In this type of diagram, the negative of a vector is the vector with the
same magnitude but in the opposite direction. Multiplying a vector by a
scalar is represented by lengthening the arrow by that factor, and similarly
for division.
Self-Check
Given vector Q represented by an arrow below, draw arrows representing the
vectors 1.5Q and —Q.
Q
Discussion Questions
A. Would it make sense to define a zero vector. Discuss what the zero
vector’s components, magnitude, and direction would be; are there any issues
here. If you wanted to disqualify such a thing from being a vector, consider
whether the system of vectors would be complete. For comparison, why is the
ordinary number system (scalars) incomplete if you leave out zero. Does the
same reasoning apply to vectors, or not.
B. You drive to your friend’s house. How does the magnitude of your
.
r vector
compare with the distance you’ve added to the car’s odometer.
7.2Calculations with Magnitude and Direction
If you ask someone where Las Vegas is compared to Los Angeles, they
are unlikely to say that the
.
x is 290 km and the
.
y is 230 km, in a coordi-
nate system where the positive x axis is east and the y axis points north.
They will probably say instead that it’s 370 km to the northeast. If they
were being precise, they might specify the direction as 38
°
counterclockwise
from east. In two dimensions, we can always specify a vector’s direction like
this, using a single angle. A magnitude plus an angle suffice to specify
everything about the vector. The following two examples show how we use
trigonometry and the Pythagorean theorem to go back and forth between
the x-y and magnitude-angle descriptions of vectors.
x
y
x
y
x component
(positive)
y component
(negative)
1.5Q–Q
Chapter 7Vectors