A Bit About Vectors for AP Physics B & C

Practice problems for these concepts can be found at: Vectors Practice Problems for AP Physics B & C

Scalars

Scalars are numbers that have a magnitude but no direction.

For example, temperature is a scalar. On a cold winter day, you might say that it is "4 degrees" outside. The units you used were "degrees." But the temperature was not oriented in a particular way; it did not have a direction.

Another scalar quantity is speed. While traveling on a highway, your car's speedometer may read "70 miles per hour." It does not matter whether you are traveling north or south, if you are going forward or in reverse: your speed is 70 miles per hour.

Vector Basics

p>Vectors, by comparison, have both magnitude and direction.

An example of a vector is velocity. Velocity, unlike speed, always has a direction. So, let's say you are traveling on the highway again at a speed of 70 miles per hour. First, define what direction is positive—we'll call north the positive direction. So, if you are going north, your velocity is +70 miles per hour. The magnitude of your velocity is "70 miles per hour," and the direction is "north."

If you turn around and travel south, your velocity is –70 miles per hour. The magnitude (the speed) is still the same, but the sign is reversed because you are traveling in the negative direction. The direction of your velocity is "south."

IMPORTANT: If the answer to a free-response question is a vector quantity, be sure to state both the magnitude and direction. However, don't use a negative sign if you can help it! Rather than "–70 miles per hour," state the true meaning of the negative sign: "70 miles per hour, south."

Graphic Representation of Vectors

Vectors are drawn as arrows. The length of the arrow corresponds to the magnitude of the vector—the longer the arrow, the greater the magnitude of the vector. The direction in which the arrow points represents the direction of the vector. Figure 9.1 shows a few examples:

Vector A has a magnitude of 3 meters. Its direction is "60° degrees above the positive x-axis." Vector B also has a magnitude of 3 meters. Its direction is "β degrees above the negative x-axis." Vector C has a magnitude of 1.5 meters. Its direction is "in the negative y-direction" or "90 degrees below the x-axis."

Vector Components

Any vector can be broken into its x- and y -components. Here's what we mean:

Place your finger at the tail of the vector in Figure 9.2 (that's the end of the vector that does not have a on it). Let's say that you want to get your finger to the head of the vector without moving diagonally. You would have to move your finger three units to the right and four units up. Therefore, the magnitude of left–right component (x -component) of the vector is "3 units" and the magnitude of up–down (y-component) of the vector is "4 units."

If your languages of choice are Greek and math, then you may prefer this explanation.

You may want to check to see that these formulas work by plugging in the values from our last example.

V_x = 5 cos 53° = 3 units

V_y = 5 sin 53° = 4 units

Some Hints

1. Make sure your calculator is set to DEGREES, not radians.
2. Always use units. Always. We mean it. Always.

Practice problems for these concepts can be found at: Vectors Practice Problems for AP Physics