What you must be able to do to succeed in AP Physics:
- Resolve a vector into 2 perpendicular components, the "X" component and the "Y" component.
- Find the resultant vector if given component vectors.
- Be able to add vectors - compute magnitude (size) and direction (angle ccw from zero degrees) of a resultant of two or more vectors.
- Know that a vector multiplied by a vector in a dot product produces a scalar. Know that calculation of the magnitude involves the cosine. (example: W = F* delta x = F*x*cos theta).
- Know that a vector times a vector in a cross product produces a vector. Know that calculation of the magnitude involves the sine. (example: F_m = qv X B)
- Be able to calculate magnitude and direction for both dot and cross products.
A scalar is a physical quantity that can be described with a single number.
A vector is a physical quantity that must be described with two numbers. The first number represents the magnitude (size) of the vector. The second number represents the direction of the vector.
Vectors that are commonly encountered in introductory physics include: displacement, velocity, acceleration, force, and field.
Vectors may be represented by an arrow drawn on a coordinate axis. You first define a scale on the paper or computer screen. The length of the arrow represents the magnitude of the vector you are representing. The angle of the arrow with the x-axis represents the vector’s direction. Angles are measured counterclockwise from zero degrees.
A vector at a non-zero angle with the x-axis can be thought of as having two components, an x-component and a y-component. The x-component is the part that goes purely in the positive or negative x-direction. The y-component is the part that goes purely in the positive or negative y-direction. When we draw the components of a vector we call it “resolving the vector into components.”
Vectors with the same direction (0 degrees to each other) are algebraically added. Vectors in opposite directions (180 degrees to each other) must be subtracted (adding the opposite). Vectors at any other angle to each other must be dealt with using the rules of vectors.
When you have properly added vectors you have found the resultant. Think of the resultant as a single vector that gets done what all the added-up vectors were trying to accomplish.
To add vectors that are not in the same or opposite directions, you may either solve the problem graphically or analytically.
In the graphical method, you draw the vectors to scale and in the proper direction on an axis. Draw the first vector from the origin, and then draw the second vector with its tail on the tip of the first vector. Repeat for as many vectors as you have, each new vector drawn with its tail on the tip of the last vector. This is called the “tip to tail method.” The resultant is the vector that extends from the origin to the tip of the last vector. Its magnitude is found by measurement with a ruler (and comparing it to the scale) and its direction is found by measurement with a protractor.
The analytical method of vector addition uses trig functions (sine, cosine, and tangent) and the Pythagorean theorem. As this deals more with mathematical procedure than with concepts, it is not discussed here. I will discuss the origin of the trig functions. The ancient Greeks were brilliant geometers. They noticed that any right triangle with the same angles had sides in the same ratios. Remember that a ratio is a division problem. Recall that right triangles have a long side, the hypotenuse. Relative to each angle they have an adjacent side and an opposite side. Imagine (or actually do it!) drawing two right triangles with the same angles, but different sides. It does not matter how much difference there is in size between the two triangles, if we divide the same two sides in each triangle (adjacent over hypotenuse, say), we get the same ratio (answer)!