Vectors & Motion

In the physics of motion, you can describe quantities in two different ways. Scalar quantities have a magnitude but no direction is associated with them. For example, distance and speed are scalar quantities. Distance is the length of the path along which an object has traveled. Speed is the distance traveled during a period of time divided by the time interval. Time, temperature, and mass are more examples of scalar quantities.

Frequently, you will need to describe the direction as well as a magnitude of a quantity. Vector quantities include a direction as well as a magnitude. Velocity and displacement are examples of vector quantities. Displacement is the length of a straight-line path from one point to another point and the direction from the first to the second point. An example of displacement between two points or positions is 35 m [N30°E] (30° east of north). A velocity might be 75 m/s [W23°N] (23° north of west).

Sometimes you have to add vectors. For example, imagine that you walked 1.3 km directly east to a friend’s house. Then, together you walked 2.6 km directly west to school. Your total displacement would be the vector sum of the two displacements. Remember, the displacement is the length of a straight-line path from the starting point to the ending point. It does not matter what happened between those two points.

The following applet allows you to experiment with adding vectors in one dimension (along a straight line). Assume that the positive direction is to the right and the negative direction is to the left. You can indicate directions with just a plus sign (+) or a minus sign (-). To use the applet, give the first and second displacement a value. Don’t forget to indicate direction. To find the total displacement, click the update button.

One Dimensional Displacement

Adding vectors becomes more complicated when the vectors are in two dimensions. The easiest way to add vectors in two dimensions is to draw them on a coordinate system. Start by placing the tail of the first vector at the origin of the coordinate system. Place the tail of the second vector at the tip of the first vector. You can find sum of the vectors, called the resultant vector, by starting at the origin and drawing a line to the tip of the second vector. You can find the magnitude or size of the resultant vector by measuring with a ruler. Find the direction of the vector with a protractor.

The applet below allows you to practice adding vectors in two dimensions. The size (magnitude) of the vector is always a positive number. In the applet, direction of the vectors is described by reporting the angle that the vector makes with the positive x- axis. To find the angle, start on the line labeled 0 degrees and rotate counterclockwise until you reach the vector.

You can also calculate the magnitude and direction of a resultant vector by using mathematical methods. For more information on solving vector addition problems, check out this website.

Two Dimensional Displacement