PHYS101: Introduction to Mechanics

To solve two-dimensional kinematic problems, we first need to understand how two-dimensional motion is represented and how to break it up into two one-dimensional components. We also need to understand vectors. A vector is a quantity that has both a magnitude (amount) and direction. Often in texts, vectors are denoted by being bolded or having a small arrow written above the vector name.

For example, a vector called A can be written as A or as $\overrightarrow{A}$. The magnitude, or amount, of the vector A equals the value of A. And, the direction of A is given by some other notation usually accompanying the value. We can think of vectors as arrows, with the length being the magnitude of the vector and the arrow pointing in the direction of the vector.

Vectors are often notated like this: $\vec{A}=A_{x}\hat{x}+A_{y}\hat{y}$. The $\hat{x}$ denotes that the magnitude $A_{x}$ is the part of the vector that protrudes the x-axis. Similarly, the $\hat{y}$ denotes that the magnitude $A_{y}$ is the part of the vector that protrudes the y-axis.

So, for example, the vector $\vec{A}=3\hat{x}+5\hat{y}$ extends down the x-axis three units while extending up the y-axis five units. This notation is called "component form" and is a preferred way of representing vectors.

Another way of representing vectors is by denoting their magnitude and direction. For example, we can denote the vector A, shown in the previous paragraph, also as 5.83 units 59 degrees from the x-axis. Notice that we need to specify that the direction has an angle with respect to the x-axis. Not only do we need an angle, but we also need a reference point from which the angle spawns. Generally, the x-axis is a convenient choice. We call this notation the magnitude-direction form.

This video discusses vector notation. Note that they use engineering notation, which replaces x-hat with i-hat and y-hat with j-hat. The meanings are the same despite these changes.