A quantity with both magnitude (size) and direction, often represented by a directed line segment (an arrow).
A useful way to think about a vector is as something that carries you from one place to another.
In a city laid out on a grid, instructions like "3 blocks east and 4 blocks north" naturally describe a vector.
On a coordinate grid, we often represent a translation using a column vector.
Definition
Column vector
A vector written in component form as $\binom{x}{y}$, representing a translation of $x$ units horizontally and $y$ units vertically.
Vectors are also frequently expressed using the standard basis vectors (or unit vectors) $\mathbf{i}=\binom{1}{0}$ and $\mathbf{j}=\binom{0}{1}$.
In this notation, $\binom{x}{y} = x\mathbf{i} + y\mathbf{j}$.
A column vector $\binom{x}{y}$ means:
move $x$ units in the $x$-direction (right if $x>0$, left if $x<0$)
move $y$ units in the $y$-direction (up if $y>0$, down if $y<0$)
Two closely related vector ideas appear constantly in geometry:
$\overrightarrow{AB}$ is the vector representing the translation (displacement) from point $A$ to point $B$.
$\overrightarrow{OA}$ is the position vector of point $A$, meaning the vector from the origin $O$ to $A$.
Definition
Position vector
The vector from the origin $O$ to a point, written $\overrightarrow{OA}$.
If $A(x_1,y_1)$ and $B(x_2,y_2)$, then $$\overrightarrow{AB}=\binom{x_2-x_1}{y_2-y_1}$$
Note
The column vector $\binom{0}{0}$ is the zero vector.
It represents "no movement" and has magnitude 0.
Vector Addition Matches Doing One Translation Then Another
When you add vectors, you combine translations.
If you first do translation $\mathbf{a}$ and then do translation $\mathbf{b}$, the overall effect is the single translation $\mathbf{a}+\mathbf{b}$.
Definition
Resultant vector
The single vector that has the same effect as performing several vectors in sequence, for example $\mathbf{a}+\mathbf{b}$.
Pictorial (Geometric) Addition: Head-to-Tail Rule
To draw $\mathbf{a}+\mathbf{b}$:
Draw $\mathbf{a}$.
From the head (arrow tip) of $\mathbf{a}$, draw a copy of $\mathbf{b}$.
The vector from the tail of $\mathbf{a}$ to the head of $\mathbf{b}$ is $\mathbf{a}+\mathbf{b}$.
Tip
If you draw $\mathbf{a}$ then $\mathbf{b}$, you get $\mathbf{a}+\mathbf{b}$.
If you draw $\mathbf{b}$ then $\mathbf{a}$, you still end at the same point, so $\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}$.
Component Addition
If $\mathbf{a}=\binom{a_x}{a_y}$ and $\mathbf{b}=\binom{b_x}{b_y}$, then $$\mathbf{a}+\mathbf{b}=\binom{a_x+b_x}{a_y+b_y}$$
This matches the idea that horizontal moves add with horizontal moves, and vertical moves add with vertical moves.
Example
Let $\mathbf{a}=\binom{3}{4}$ and $\mathbf{b}=\binom{-1}{2}$.
Then $$\mathbf{a}+\mathbf{b}=\binom{3+(-1)}{4+2}=\binom{2}{6}$$
So the combined translation is 2 units right and 6 units up.
Vector Subtraction Means Adding the Opposite Direction
Subtracting a vector means undoing it.
Definition
Negative of a vector
For a vector $\mathbf{a}$, the vector $-\mathbf{a}$ has the same magnitude but the opposite direction.
So $$\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})$$
Geometrically, $-\mathbf{b}$ is $\mathbf{b}$ reversed.
Common Mistake
A common mistake is to treat vectors like ordinary numbers and "cancel" without thinking about direction.
Always interpret $\mathbf{a}-\mathbf{b}$ as "do $\mathbf{a}$, then do the reverse of $\mathbf{b}$".
Displacement Between Two Points Using Position Vectors
If you know position vectors $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$, then the displacement from $A$ to $B$ is $$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\mathbf{b}-\mathbf{a}$$
Example
If $\overrightarrow{OA}=\binom{1}{2}$ and $\overrightarrow{OB}=\binom{5}{-1}$, then $$\overrightarrow{AB}=\binom{5-1}{-1-2}=\binom{4}{-3}$$
So from $A$ to $B$ you go 4 right and 3 down.
Scalars Stretch or Shrink Vectors (and Can Reverse Them)
A scalar is an ordinary number. Multiplying a vector by a scalar changes its length.
Definition
Scalar multiple
A vector of the form $k\mathbf{a}$, where $k$ is a number (scalar). It is parallel to $\mathbf{a}$ and has magnitude $|k|\,|\mathbf{a}|$.
If $k>0$, the direction stays the same.
If $k<0$, the direction reverses.
In component form, $$k\binom{x}{y}=\binom{kx}{ky}$$
Example
If $\mathbf{a}=\binom{2}{-3}$ then $3\mathbf{a}=\binom{6}{-9}$ and $-\tfrac12\mathbf{a}=\binom{-1}{\tfrac32}$.
Note
Thinking "a column vector describes a translation" makes scalar multiples intuitive: $2\mathbf{a}$ means doing the same translation twice in the same direction.
Midpoints and "Go Via a Point" Strategies Make Geometry Easier
Vector methods become especially efficient when points divide segments in known ratios.
Midpoint as a Vector Relationship
If $M$ is the midpoint of $AB$, then $$\overrightarrow{AM}=\tfrac12\overrightarrow{AB} \quad\text{and}\quad \overrightarrow{OM}=\tfrac12(\overrightarrow{OA}+\overrightarrow{OB})$$
Note
This matches a key idea: if $\overrightarrow{AM}=\mathbf{a}$ and $M$ is the midpoint, then $\overrightarrow{AB}=2\mathbf{a}$.
Example
$M$ is midpoint of $AB$, and $\overrightarrow{OA}=3\mathbf{a}$, $\overrightarrow{OB}=2\mathbf{b}$.
Then $$\overrightarrow{OM}=\tfrac12(3\mathbf{a}+2\mathbf{b})$$
If $\overrightarrow{ON}=2\overrightarrow{OM}$, then $\overrightarrow{ON}=3\mathbf{a}+2\mathbf{b}$ and $$\overrightarrow{AN}=\overrightarrow{ON}-\overrightarrow{OA}=(3\mathbf{a}+2\mathbf{b})-3\mathbf{a}=2\mathbf{b}$$
The "Go Via a Point" Rule
When a vector is not directly given, you can often split the journey through a convenient point (often $A$ or the origin $O$).
For instance, $$\overrightarrow{MN}=\overrightarrow{MA}+\overrightarrow{AN}$$
This is the same reasoning as "to go from $M$ to $N$, go from $M$ to $A$, then from $A$ to $N$".
Example
In triangle $ABC$, $M$ is midpoint of $AB$ and $N$ is midpoint of $AC$.
Let $\overrightarrow{AM}=\mathbf{a}$ and $\overrightarrow{AN}=\mathbf{b}$.
Then $\overrightarrow{AB}=2\mathbf{a}$ and $\overrightarrow{AC}=2\mathbf{b}$.
And $$\overrightarrow{BC}=\overrightarrow{BA}+\overrightarrow{AC}=-2\mathbf{a}+2\mathbf{b}=2(\mathbf{b}-\mathbf{a})$$
So $\overrightarrow{BC}$ is a scalar multiple of $\overrightarrow{MN}$, therefore $MN \parallel BC$, meaning $MNCB$ is a trapezoid.
Vector Expressions Can Prove Parallel Lines and Parallelograms
A major strength of vectors is that they turn geometric properties into algebraic statements.
Parallelism Through Scalar Multiples
Two non-zero vectors are parallel if one is a scalar multiple of the other: $$\mathbf{u} \parallel \mathbf{v} \iff \mathbf{u}=k\mathbf{v}\ \text{for some scalar }k$$
Definition
Parallel vectors
Two vectors that point in the same or opposite direction. Equivalently, one is a scalar multiple of the other.
Parallelogram Test Using Opposite Sides
A quadrilateral $ABCD$ is a parallelogram if opposite sides are equal as vectors, for example $$\overrightarrow{AB}=\overrightarrow{DC} \quad\text{and}\quad \overrightarrow{BC}=\overrightarrow{AD}$$
Example
Four points have position vectors: $$\overrightarrow{OA}=\mathbf{a}+\mathbf{b}$$ $$\overrightarrow{OB}=\mathbf{a}-\mathbf{b}$$ $$\overrightarrow{OC}=-\mathbf{a}-\mathbf{b}$$ $$\overrightarrow{OD}=-\mathbf{a}+\mathbf{b}$$ (where $\mathbf{a}$ and $\mathbf{b}$ are non-zero and not parallel).
Then $$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=(\mathbf{a}-\mathbf{b})-(\mathbf{a}+\mathbf{b})=-2\mathbf{b}$$
And $$\overrightarrow{DC}=\overrightarrow{OC}-\overrightarrow{OD}=(-\mathbf{a}-\mathbf{b})-(-\mathbf{a}+\mathbf{b})=-2\mathbf{b}$$
So $\overrightarrow{AB}=\overrightarrow{DC}$, meaning opposite sides are equal and parallel.
Similarly, you can show $\overrightarrow{BC}=\overrightarrow{AD}$, so $ABCD$ is a parallelogram.
Note
Stating that $\mathbf{a}$ and $\mathbf{b}$ are non-zero and not parallel prevents degenerate cases (for example, points collapsing so the "parallelogram" has zero area).
Solving for Unknown Vectors in a Diagram
Often you are given several displacements and must find another by choosing a route.