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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