There are two ways that we represent vectors in physics classes. The first is a geometric representation and the second is an algebraic one. The geometric representation is great for building up intuition and an understandig of certain concepts, but it’s not so good for actually performing calculations. The algebraic representation is great for doing calculations, but it’s easy to lose track of the big picture when working with them. In the end, you will want to be familiar with both perspectives.

The most natural way to geometrically capture the concepts of direction and magnitude in a picture is by drawing an arrow. If we think about the arrow as an instruction that tells us which way to push an object, we can match our push so that it goes in the same direction as the arrow, and then the length of the arrow is an indicator of how hard to push. A short arrow is a light push and a long arrow is a strong push. We can also translate this concept into the velocity of an object. The arrow says that the object is moving in that direction and we use its length to indicate how fast it’s going.

We call the starting point of the arrow the tail of the vector, and the tip of the arrow the head of the vector. It’s important to note that the vector itself does not have a location. It only tells us about the relative positions of the starting and ending locations of the vector. In other words, we can slide the vector around the plane (without rotating it) and it doesn’t change the vector itself.

The algebraic notation feels a lot like how we plot points in the \(xy\) plane. A vector that points \(x\) spaces to the right and \(y\) spaces up is denoted by \(\langle x, y \rangle\text{.}\) Notice that the brackets are pointed rather than being round. This small shift in notation gives is what indicates the difference between a vector and a point. The \(x\) and \(y\) values are called the components of the vector (or vector components). More specifically, if the diagram is oriented in the "traditional" manner, the first component corresponds to the part of the vector that points in the horizontal direction, and the second component corresponds to the part of the vector that points in the vertical direction. There are more complicated ways to interpret the components of a vector, but we won’t be needing them for this class.

If draw the vector with its tail at the origin, we say that the vector is in standard position. In this case, the algebraic notation for the vector "matches" with the coordinates of the head. That is, when the vector \(\langle x, y \rangle\) is drawn in standard position, the head of the vector will be located at the point \((x,y)\text{.}\) However, this only happens when the vector is in standard position. If the tail is somewhere else, then the head of the vector is found relative to the tail, and it will no longer correspond to the point.

The geometric picture of vector addition is to think about a triangle. If we are given two vectors, you draw them "tail-to-head" often intuitively framed as combining movements back-to-back. In other words, we think of the vectors as representing motion, and follow the motions in sequence. This is precisely what position vectors do for us, but it’s not necessarily the best image if the vectors represented something else, like forces or electric fields. In those cases, the core concept isn’t motion, and so this doesn’t create the best picture. So the more general way of thinking about geometric vector addition is known as the method of adding vectors. This accomplishes the same thing, but avoids the concept of "movement" in the process.

Algebraically, vector addition is accomplished by performing "component-wise" addition. That is, if we wanted to calculate \(\langle x_1, y_1 \rangle + \langle x_2, y_2 \rangle \text{,}\) the result is attained by adding the components that are in the same position, which gives \(\langle x_1 + x_2, y_1 + y_2 \rangle \text{.}\) The good news is that this is what most students would naturally end up doing. It’s not hard to see that this gives the same result as the geometric method because the horizontal and vertical directions are independent of each other.

Multiplying a vector by a scalar (a number) results in stretching the vector by the scalar value and flipping the direction if the scalar is negative. Geometrically, this is straight-forward. Algebraically, this is accomplished by multiplying each of the components by the scalar. It turns out that the underlying mathematics is nothing more than the distributive property, but those are details that we don’t need to get into here.

One of the most important ideas to keep in mind with vectors is that (at least in physics) they always exist in a specific context. In other words, the vectors mean something about the physical world, and so when we add vectors or multiply vectors by a scalar, it will correspond to some sort of physical outcome. Without going into detail, here are some examples.