2 Chapter 2: Vectors
Link to online textbook chapter
Khan Academy: Videos about Vectors
Section 2.1: Scalars vs. Vectors
Link to Section 2.1 of the textbook
A scalar is something you can measure. You have 45 bananas, or it’s 72 degrees in the room, or your tire has 3 holes in it. These are all scalar quantities. Common examples of scalars we will see in this class are distance, speed, temperature, energy, and length. Scalars can be positive or negative, but we’ll usually just see the positive aspect of them in class.
A vector has both term for how much you have of something (called a magnitude) and a direction (where you’re going). Common examples of vectors we’ll see in this class are displacement, velocity, acceleration, and forces.
Vectors are represented by either a line over a variable, , a bold variable, , or a line under a variable, . In physics, we use the notation for vectors, except where the format doesn’t support that option, in which case we’ll use a bold value.
The magnitude of a vector is found by removing the directional component. A magnitude is always a positive value. The following notation means to take the magnitude of the vector: . Note that a vector without direction is simply a scalar value .
Section 2.2: Graphical Addition of Vectors
Link to Section 2.3 of the textbook
Section 2.3: Component Vectors
Link to Section 2.2 of the textbook
Any vector can be written as the sum of two vectors in any given coordinate system.
In the figure on the right, the vector can be described as the sum of the vectors and . Why would we do this? We are breaking up the vector into its component vectors in the coordinate system. lies along the -axis, and lies along the -axis. We can describe a vector pointing in any two dimensional direction using the coordinate system in this manner.
We can now write . Let’s recall some basic geometry. In a right triangle, the Sine of an angle is the side of the triangle Opposite the angle divided by the Hypotenuse of the triangle (SOH). The sine of the angle in the figure above is given by sin() = . The Cosine of an angle is given by the Adjacent side to the angle divided by the Hypotenuse. The cosine of is given by cos() = (CAH). The Tangent of the angle is found by dividing the Opposite side by the Adjacent side, tan() = (TOA). Thus, the abbreviation SOH CAH TOA.
Note that the system above depends on which angle you call . Every time you try to solve a vector graphically in this manner, you must draw a picture and indicate which angle is which.
By rearranging the above equations, we can come to the conclusion that sin() and cos(). Now we can write the vector as the sum of its component vectors expressed as functions of , sin() + cos().
The figure to the left illustrates the components of the sum of vectors . Vectors , , and have been broken into their components, where , , and . Vectors and have been slightly offset from their positions for clarity.
Above, . However, since , and , we can also write this as