Vectors Describe Translations and Displacements
Vector
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.
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 $A$.
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 $$\overrightarrow{AB}=\binom{x_2-x_1}{y_2-y_1}$$
- 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 $\mathbf{a}+\mathbf{b}$.
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 $\mathbf{a}+\mathbf{b}$.
- 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.
- 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.
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.
- 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 $$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}=\mathbf{b}-\mathbf{a}$$
- 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.
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, $$k\binom{x}{y}=\binom{kx}{ky}$$
If $\mathbf{a}=\binom{2}{-3}$ then $3\mathbf{a}=\binom{6}{-9}$ and $-\tfrac12\mathbf{a}=\binom{-1}{\tfrac32}$.
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 $$\overrightarrow{AM}=\tfrac12\overrightarrow{AB} \quad\text{and}\quad \overrightarrow{OM}=\tfrac12(\overrightarrow{OA}+\overrightarrow{OB})$$
This matches a key idea: if $\overrightarrow{AM}=\mathbf{a}$ and $M$ is the midpoint, then $\overrightarrow{AB}=2\mathbf{a}$.
- $M$ 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 $N$".
- 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 $\overrightarrow{AC}=2\mathbf{b}$.
- Also, $$\overrightarrow{MN}=\overrightarrow{MA}+\overrightarrow{AN}=-\mathbf{a}+\mathbf{b}=\mathbf{b}-\mathbf{a}$$
- 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: $$\mathbf{u} \parallel \mathbf{v} \iff \mathbf{u}=k\mathbf{v}\ \text{for some scalar }k$$
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 $$\overrightarrow{AB}=\overrightarrow{DC} \quad\text{and}\quad \overrightarrow{BC}=\overrightarrow{AD}$$
- 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.
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.
- Given: $$\overrightarrow{AD}=\mathbf{a}+\mathbf{b}$$ $$\overrightarrow{BD}=2\mathbf{b}-\mathbf{a}$$ $$ \overrightarrow{BC}=3\mathbf{a}$$
- To find $\overrightarrow{AB}$:
- Go from $A$ to $B$ via $D$: $$\overrightarrow{AB}=\overrightarrow{AD}+\overrightarrow{DB}$$
- But $\overrightarrow{DB}=-\overrightarrow{BD}=-(2\mathbf{b}-\mathbf{a})=\mathbf{a}-2\mathbf{b}$.
- So $$\overrightarrow{AB}=(\mathbf{a}+\mathbf{b})+(\mathbf{a}-2\mathbf{b})=2\mathbf{a}-\mathbf{b}$$
- To find $\overrightarrow{DC}$:
- Go from $D$ to $C$ via $B$:
- $$\overrightarrow{DC}=\overrightarrow{DB}+\overrightarrow{BC}=(\mathbf{a}-2\mathbf{b})+3\mathbf{a}=4\mathbf{a}-2\mathbf{b}=2(2\mathbf{a}-\mathbf{b})$$
- So $\overrightarrow{DC}$ is a scalar multiple of $\overrightarrow{AB}$, therefore $AB \parallel DC$.
- What translation does $\binom{-2}{5}$ represent?
- If $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$, what is $\overrightarrow{BA}$ in terms of $\mathbf{a},\mathbf{b}$?
- If $M$ is midpoint of $AB$, write $\overrightarrow{OM}$ in terms of $\overrightarrow{OA}$ and $\overrightarrow{OB}$.
