Vectors, Dot Products, and Cross Products

When studying the flight of an airplane, a rocket firing, or countless other applications, we can use vectors to mathematically describe what's going on. A vector has two components: magnitude and direction. This is similar to the use of polar coordinates, where we describe a point by its distance from the origin (like the magnitude of a vector) and the angle formed by the line from the origin to the point (like the direction of a vector).

Since all that distinguishes a vector from other vectors is its magnitude and direction, we can move it around at will. Therefore, for the sake of convenience, we often place the tail of the vector at the origin and describe the vector by the location of the head. This way of describing a vector looks just like describing a point, so to be clear that we're talking about a vector (that when the tail is placed at the origin has its head at this location), we use angled brackets, like so: v⃗ =⟨1,3,5⟩ This vector is shown below.