155

Addition of vectors given their magnitudes and directions

In this case, you must first translate the magnitudes and directions into

components, and the add the components.

Graphical addition of vectors

Often the easiest way to add vectors is by making a scale drawing on a

piece of paper. This is known as graphical addition, as opposed to the

analytic techniques discussed previously.

Example

Question: Given the magnitudes and angles of the

.

r vectors

from San Diego to Los Angeles and from Los Angeles to Las

Vegas, find the magnitude and angle of the

.

r vector from San

Diego to Las Vegas.

Solution: Using a protractor and a ruler, we make a careful scale

drawing, as shown in the figure. A scale of 1 cm

.

100 km was

chosen for this solution. With a ruler, we measure the distance

from San Diego to Las Vegas to be 3.8 cm, which corresponds to

380 km. With a protractor, we measure the angle

.

to be 71

°

.

Even when we don’t intend to do an actual graphical calculation with a

ruler and protractor, it can be convenient to diagram the addition of vectors

in this way. With

.

r vectors, it intuitively makes sense to lay the vectors tip-

to-tail and draw the sum vector from the tail of the first vector to the tip of

the second vector. We can do the same when adding other vectors such as

force vectors.

Self-Check

How would you subtract vectors graphically.

Discussion Questions

A. If you’re doing graphical addition of vectors, does it matter which vector you

start with and which vector you start from the other vector’s tip.

B. If you add a vector with magnitude 1 to a vector of magnitude 2, what

magnitudes are possible for the vector sum.

C. Which of these examples of vector addition are correct, and which are

incorrect.

Los

Angeles

Las Vegas

San Diego

190 km

370 km

141

°

38

°

distance=.

.

=.

Section 7.3Techniques for Adding Vectors

A

B

A+B

AB

A+B

A

B

A+B

B

A

A+

B

A

B

Vectors can be added graphically by

placing them tip to tail, and then

drawing a vector from the tail of the

first vector to the tip of the second

vector.

The difference A–B is equivalent to A+(–B), which can be calculated graphically by reversing B to form –B, and

then adding it to A.