Vectors

Vectors are lines that have a size and point in a specific direction. The size of a vector is called its magnitude. In this chapter, you’ll review an introduction to vectors, including their notation and how to add or subtract them.

Vector notation

Vectors are described using different variables from the usual ones you’ve seen. Instead of using $x$ and $y$ directly, we use $i$ for the $x$ direction and $j$ for the $y$ direction. These are written as algebraic terms, not as coordinate points.

• $i$ measures how far the vector moves in the $x$ direction.
• $j$ measures how far the vector moves in the $y$ direction.

So, for a vector that begins at $(0,0)$ and ends at $(1,2)$, the change in $x$ is $1$ and the change in $y$ is $2$. In vector notation, that’s $i+2j$. This tells you the vector moves one unit to the right ($i$) and two units up ($2j$).

Try turning the following examples into vector notation:

A vector begins at $(0,0)$ and ends at $(-2,3)$

A vector begins at $(3,0)$ and ends at $(0,0)$

A vector begins at $(5,2)$ and ends at $(7,7)$

Vector addition

Adding by algebra

When you’re given multiple vectors, you can add them by treating $i$ and $j$ like variables and combining like terms.

• Add the $i$ terms together.
• Add the $j$ terms together.

Look at the following example:

What is the sum of the following vectors?

The first vector goes from $(0,0)$ to $(2,2)$. That’s an increase in $x$ of $2$ and an increase in $y$ of $2$, so in vector notation it’s $2i+2j$.

The second vector goes from $(2,-1)$ to $(2,1)$. There is no change in $x$, and the increase in $y$ is $2$, so in vector notation it’s $2j$. (There is no $i$ term because the $x$ position doesn’t change.)

To find the sum, add the like components:

$$\begin{array}{rl}& =2i+2j+0i+2j\\ & =2i+4j\end{array}$$

So, the sum of the two vectors is $2i+4j$.

Adding by visualization

To see the vector that results from adding two vectors, connect them head to tail. Then draw a new vector from the starting point to the new ending point. That new vector is the sum of the original vectors.

If $a$ and $b$ are the same vectors used in the previous example, the new vector goes from $(0,0)$ to $(2,4)$, giving the vector $2i+4j$. This matches the result from adding the expressions.

Vector subtraction

Subtracting by algebra

Subtracting vectors works like addition, except you subtract the components.

• Subtract the $i$ terms.
• Subtract the $j$ terms.

The first vector goes from $(1,0)$ to $(0,2)$. That’s a decrease in $x$ of $1$ and an increase in $y$ of $2$, so in vector notation it’s $-i+2j$.

The second vector goes from $(0,-2)$ to $(2,0)$. That’s an increase in $x$ of $2$ and an increase in $y$ of $2$, so in vector notation it’s $2i+2j$.

To find the difference, subtract the like components of the second vector from the first:

$$\begin{array}{rl}& =-i+2j-(2i+2j)\\ & =-3i\end{array}$$

So, the difference of the two vectors is $-3i$.

Subtracting by visualization

To subtract one vector from another, flip the vector being subtracted (so it points in the opposite direction). Then connect the vectors head to tail, just like in addition. Finally, draw a new vector from the starting point to the new ending point. That new vector is the difference of the original vectors.