Vectors

In physics, we quantify most things with either a scalar or a vector. A scalar is described by one number; mass, temperature, and time are scalars. A vector is said to have both magnitude and direction; displacement, velocity, acceleration, and force are vectors. Note that if you ever study linear algebra, there are more rigorous definitions for these mathematical objects.

But, for now, let’s turn to a familiar one-dimensional vector: the displacement of a bunny along a number line. When you first learned arithmetic, a bunny hopped along a number line: $3 - 5$ is evaluated by starting at $0$, hopping $3$ to the right, then hopping $5$ to the left, and landing on $-2$. Each hop has a magnitude (how far) and a direction (right for positive, left for negative).

Give the bunny a second way to hop — up and down as well as left and right — and we now have a two-dimensional “hop vector”.

Components

To work with vectors algebraically, we must express the individual components of each vector. In two dimensions, we may write a vector as:

The magnitude, i.e. the length of the arrow representation shown below, can be found using the Pythagorean theorem:

The direction angle is then (usually) measured counterclockwise from the positive $x$ axis:

To work with vectors graphically, we draw them using an arrow. The starting point is often called the “tail” of the vector, and the ending point is called the “tip” (arrowhead). The length of the arrow represents the vector’s magnitude, while the direction is captured by the orientation of the arrow.

Addition

Adding vectors is simply letting the bunny hop multiple times, and then looking at the combined effect of the hops. For example, in evaluating $\vec{a} + \vec{b}$, think of $\vec{a}$ as hop 1 and $\vec{b}$ as hop 2. To find where the bunny ends up, place the tail of hop 2 at the tip of hop 1. The sum points from the original tail to the final tip. This is called tip-to-tail addition.

Symbolically, add matching components:

The order of the hops does not matter. Hop 1 followed by hop 2 lands in exactly the same place as hop 2 followed by hop 1:

We can continue adding any number of hops in this fashion, joining them tip to tail. In the case of our rabbit, the total vector sum represents the net displacement from the starting location.