Question 1.6: Finding components of vectors Here we will use vector compon...
Finding components of vectors Here we will use vector components to track the positions of Raoul and Maria as they set out walking from their aunt’s house. For simplicity, we will assume that the aunt’s house is at the origin of our coordinate system. Raoul walks a certain distance straight east and then a certain distance straight north. He ends up at a position that is 500 m from the house in a direction that is 35°north of east. Maria, instead, walks first straight west and then straight south. Her final position is 700 m from the starting point in a direction that is 55° south of west. How far did Raoul walk on the east leg of his trip? How far on the north leg? Similarly, how far did Maria walk on the west leg and how far on the south leg? Express these quantities in terms of displacement vectors and their components, using a coordinate system with the +x axis pointing east and the +y axis pointing north.
SET UP First we sketch a diagram showing each displacement, as in Figure 1.18. We draw a rectangular coordinate system and label the axes, with the +x axis east and the +y axis north. Be sure to make your sketch large enough so that you will be able to draw and label the components of each vector. Note that the vector representing Maria’s displacement lies in the third quadrant, so both the x and y components of this vector will be negative.
SOLVE We’ll label Raoul’s displacement vector \overset{\rightarrow }{\pmb{A}} and Maria’s \overset{\rightarrow }{\pmb{B}} .
Then to find the components, we use Equations 1.2:
A_x=A\cos\theta and B_x=B=\cos \theta
A_y=A\sin \theta B_y=B\sin \theta
We measure the angles counterclockwise from the +x axis. So the direction of Maria’s displacement is 180° + 55° = 235°. The components of the two displacement vectors are as follows:
| Raoul | Maria |
| A_x=500 m(\cos35°)
= 410 m (east) |
B_x=700 m(\cos235°)
= -402 m (west) |
| A_y=500 m(\sin35°)
= 287 m (north) |
B_y=700 m(\sin235°)
= -573 m (south) |
REFLECT Because Maria walked west and south, the x and y components of her displacement are both negative. To get the correct signs, we must measure the angles counterclockwise from the +x axis.
Practice Problem: Raoul’s friend Johnny sets out from his house in the city. His displacement is 600 m 40° north of west. What are the x and y components of his displacement vector? Answers: A_x = -460 m; A_y = 386 m.
