3.2 Using Vectors

Suppose Sam starts from his front door, walks across the street, and ends up 200 ft to the northeast of where he started. Sam’s displacement, which we will label $\vec{S}$ , is shown in FIGURE 3.2a. The displacement vector is a straight-line connection from his initial to his final position, not necessarily his actual path.

The figure consists of two parts labeled ("a") and (b) that illustrate the movement of Sam and Bill using displacement vectors.

FIGURE 3.2 Full Alternative Text

To describe a vector we must specify both its magnitude and its direction. We can write Sam’s displacement as $\vec{S} = (200 ft, northeast)$ . The magnitude of Sam’s displacement is $S = | \vec{S} | = 200 ft,$ the distance between his initial and final points.

Sam’s next-door neighbor Bill also walks 200 ft to the northeast, starting from his own front door. Bill’s displacement $\vec{B} = (200 ft, northeast)$ has the same magnitude and direction as Sam’s displacement $\vec{S}$ . Because vectors are defined by their magnitude and direction, two vectors are equal if they have the same magnitude and direction. Thus the two displacements in FIGURE 3.2b are equal to each other, and we can write $\vec{B} = \vec{S}$ .

NOTE A vector is unchanged if you move it to a different point on the page as long as you don’t change its length or the direction it points.