Adding vectors

We have described a mathematical structure that allows us to code both direction and magnitude of a quantity — vectors. One way to think of it is as an arrow on a coordinate system starting from the origin and going off in a particular direction for a certain distance. The basic physical system we map onto this mathematical system is the displacement of an object in space.

A mental model for adding vectors

This physical model gives us a quick and easy way to think about how to add two vectors. You start at the origin and traverse the first vector from tail to head. (You undergo the first displacement.) Then, from where you are, you traverse the second arrow from tail to head. (You undergo the second displacement.) The result is the arrow drawn from the tail of the first arrow directly to the head of the second. (Your overall displacement.)

This sense of meaning of vector addition directly gives us a graphical way to add vectors. For example,  if we were consider displacement in a 2D space described by a "graph for the eye" that is an x-y plane, and we had two displacements labeled $\overrightarrow{r_1}$  (shown in red) and $\overrightarrow{r_2}$ (shown in blue),  then the sum of these two displacements, $\overrightarrow{r}$ (shown in black), is what you get by doing one displacement after the other as shown in the figure:

$\overrightarrow{r} =$ $\overrightarrow{r_1}$ + $\overrightarrow{r_2}$

This provides a graphical method for adding two vectors.

An algebraic way to add vectors

We also showed in our discussion of vectors how we could represent them algebraically using the unit direction vectors $\hat{i}$ and $\hat{j}$. A general 2D vector can be represented by the sum of displacements in two perpendicular directions: $x$ and $y$. An arbitrary vector (often written as $(x,y)$ in math class) that has a displacement of the amount $x$ in the $x$ direction and an amount $y$ in the $y$ direction can be written

$$\overrightarrow{r} = x\hat{i} + y\hat{j}$$

The values $x$ and $y$ are referred to as the components of $\overrightarrow{r}$. 

If we have two such vectors and want to add them, we can do so using straightforward algebra: rearranging and regrouping.

$$\overrightarrow{r_1} = x_1\hat{i} + y_1\hat{j}$$

$$\overrightarrow{r_2} = x_2\hat{i} + y_2\hat{j}$$

$$\overrightarrow{r} = \overrightarrow{r_1} + \overrightarrow{r_2}$$

Replacing $r_1$ and $r_2$ by their component forms:

$$\overrightarrow{r} =(x_1\hat{i} + y_1\hat{j}) + ( x_2\hat{i} + y_2\hat{j})$$

and then regrouping

$$\overrightarrow{r} =(x_1\ + x_2)\hat{i} + (y_1 +  y_2)\hat{j}$$

If we then identify the sum, r, as having components $x$ and $y$, we can identify what they are:

$$\overrightarrow{r} = \overrightarrow{r_1} + \overrightarrow{r_2} = x\hat{i} + y\hat{j}$$

$$x = x_1 + x_2$$

$$y = y_1 + y_2$$

So this gives the satisfying result that to add two vectors, we just add the $x$ components and call it the new $x$ component, and add the $y$ components and call it the new $y$ component.

A mathematician's way of adding vectors

A third way of adding vectors is natural and appropriate when you are considering the properties of the set of vectors as a vector space — a mathematical structure that gives insight into the mathematical properties of vectors. In this case, all vectors are moved so that they start at the origin. Then, to construct the sum of two vectors, you build an imaginary parallelogram using the two given vectors as two of the sides. The sum is then the diagonal of the parallelogram (starting at the origin). You can see how this is done in the figure below. 

You can see the relation of these in various forms of vector addition methods that you may encounter. Drawing these diagrams carefully can help substantially in figuring out what to do and generating the correct answer.

Subtracting vectors

Subtracting vectors is no more difficult than adding vectors, since vector math follows the standard rules of algebra. Subtracting a vector is the same as adding the negative of the vector. This is like in standard algebra, where -2 = +(-2). For example

$$\overrightarrow{a} = a_x\hat{i} + a_y\hat{j}$$

$$\overrightarrow{b} = b_x\hat{i} + b_y\hat{j}$$

$$\overrightarrow{a} - \overrightarrow{b}  = \overrightarrow{a} + (- \overrightarrow{b}) = a_1\hat{i} + a_1\hat{j} + \big( -b_x\hat{i} - b_y\hat{j}\big)$$

$$\overrightarrow{a} - \overrightarrow{b}  = \big(a_x - b_x\big)\hat{i} + \big(a_y - b_y\big)\hat{j}$$

In our geometrical diagrams, subtracting is the same as adding the reversed vector.

See the examples in the Follow-ons to see vector addition and subtraction worked out in specific cases.

Workout: Adding multiple vectors

Joe Redish 10/4/13