Vector Notation and Representation
Vectors live in a world of symbols. You'll encounter $\vec{a}$, $\mathbf{a}$, $\underline{a}$, and $\vec{AB}$ all meaning the same thing — but only if you know the rules. This lesson fixes the notation so that the rest of Module 6 is frictionless. You'll also learn why you can freely move an arrow around a diagram without changing the vector it represents.
You see $\vec{AB}$ written on an exam paper. Without looking it up — what three pieces of information does this symbol carry? Also, how is the notation $\vec{AB}$ different from $|\vec{AB}|$?
Vectors are written in multiple ways depending on the context. You must recognise and use all of them.
The four main notations for a vector $a$:
- Arrow notation: $\vec{a}$ (typed) or $\overrightarrow{AB}$ (from point $A$ to $B$)
- Bold type: $\mathbf{a}$ (used in textbooks)
- Underline: $\underline{a}$ (standard for handwriting in HSC)
- Two-point form: $\vec{AB}$ — starts at tail $A$, ends at head $B$
The magnitude (length) is written $|\vec{a}|$, $|\mathbf{a}|$, or simply $a$ (italicised, no decoration).
Key facts
- Four notations for vectors: $\vec{a}$, $\mathbf{a}$, $\underline{a}$, $\vec{AB}$
- Magnitude is written $|\vec{a}|$ or $a$ (scalar, always $\geq 0$)
- $\vec{BA} = -\vec{AB}$ (reversing points reverses direction)
- Vectors are "free" — they can be translated without changing identity
Concepts
- Why the same vector can appear anywhere on a diagram
- The geometric meaning of the zero vector $\vec{0}$
- The distinction between $\vec{AB}$ and $|\vec{AB}|$
Skills
- Read and write vectors using all four notations correctly
- Identify equal vectors in a geometric figure
- State the negative of a given vector $\vec{AB}$
You must be comfortable with all four notations used for vectors in the HSC:
In typed work (exams, textbooks): use $\vec{a}$ or $\mathbf{a}$.
In handwriting: use $\underline{a}$ (underline) — this is the HSC convention. The arrow or bold is impossible to reproduce clearly by hand.
The two-point form $\vec{AB}$: this notation tells you the tail ($A$) and the head ($B$). The three pieces of information in $\vec{AB}$ are:
- The starting point (tail): $A$
- The ending point (head): $B$
- The direction: from $A$ toward $B$ (the arrowhead points at $B$)
The magnitude of $\vec{AB}$ is $|\vec{AB}|$ or simply the length $AB$ (without arrow, without bold).
Four notations: (1) bold $\mathbf{a}$; (2) tilde $\underset{\sim}{a}$; (3) arrow $\vec{a}$; (4) directed segment $\overrightarrow{AB}$. Equal vectors: same magnitude and direction.
Pause — copy all four HSC notations for the same vector: bold, tilde, arrow, and directed segment notation — and note which the exam typically uses into your book.
Quick check: In the vector $\vec{PQ}$, which point is the tail?
We just saw the four HSC vector notations: bold $\mathbf{a}$, tilde $\underset{\sim}{a}$, arrow $\vec{a}$, and directed segment $\overrightarrow{AB}$. That raises a question: geometrically, a vector is a directed line segment (an arrow), but a vector can be translated anywhere in the plane — what does it mean for two arrows to represent the same vector? This card answers it → two directed segments represent the same vector if and only if they have equal length and equal direction (they are parallel and of the same orientation).
Geometrically, every vector is represented by a directed line segment — an arrow from its tail to its head. The length of the arrow is the magnitude; the direction the arrow points is the vector's direction.
Crucially, vectors are free: you can translate (slide) an arrow anywhere on the plane without changing the vector. Two arrows represent the same vector if and only if they have the same length and point in the same direction.
This is exactly what happens in a parallelogram: $ABCD$ is a parallelogram iff $\vec{AB} = \vec{DC}$ (note the order of letters).
Geometrically, every vector is represented by a directed line segment — an arrow from its tail to its head. The length of the arrow is the magnitude; the direction the arrow points is the vector's direction.
Pause — copy the free-vector principle: two directed segments are equal vectors iff equal magnitude AND equal direction; include a diagram with three equal vectors in different positions into your book.
Did you get this? True or false: Moving a vector arrow to a different position on the diagram changes the vector.
Worked examples · 3 in a row, reveal as you go
Write the vector from point $M$ to point $N$ in all four notations. State its magnitude.
In parallelogram $ABCD$, identify three pairs of equal vectors formed by the sides and diagonals.
Given $\vec{PQ}$, express $\vec{QP}$ in terms of $\vec{PQ}$. Explain geometrically.
Fill the gap: In parallelogram $ABCD$, $\vec{AB} = $ and $\vec{BA} = $ .
Misconceptions to fix · the 3 traps that cost marks
Did you get this? True or false: In parallelogram $ABCD$, $\vec{AB} = \vec{CD}$.
Activities · practice with the ideas
Write the vector from $X$ to $Y$ in all four notations, naming the single-letter vector $\mathbf{v}$.
In rectangle $PQRS$ (vertices in order), state three pairs of equal vectors formed by the sides.
Given $\vec{u} = \vec{AB}$, express $\vec{BA}$ in terms of $\vec{u}$, and verify that $\vec{AB} + \vec{BA} = \vec{0}$.
Explain in one sentence why an arrow representing a vector can be freely moved to any position in the plane.
If $|\vec{PQ}| = 7$ cm, what is $|\vec{QP}|$? Justify your answer.
Odd one out: Three of these statements are true. Which one is FALSE?
Earlier you were asked what three pieces of information $\vec{AB}$ carries.
The answer: (1) the tail $A$, (2) the head $B$, and (3) the direction from $A$ to $B$. The magnitude is captured implicitly by the length of the line segment from $A$ to $B$. And $|\vec{AB}|$ strips the direction away, leaving only the scalar length.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. Write the vector from point $C$ to point $D$ using all four standard notations (name the vector $\mathbf{w}$). (1 mark)
Q2. In parallelogram $WXYZ$ (vertices in order), state which vector equals $\vec{WX}$ and explain why $\vec{WX} \neq \vec{XY}$. (2 marks)
Q3. Explain why $|\vec{AB}| = |\vec{BA}|$ even though $\vec{AB} \neq \vec{BA}$. (2 marks)
Comprehensive answers (click to reveal)
Activity answers: 1. $\vec{XY}$ (two-point), $\vec{v}$ (typed), $\mathbf{v}$ (print), $\underline{v}$ (handwriting). · 2. $\vec{PQ} = \vec{SR}$; $\vec{QR} = \vec{PS}$; plus diagonal midpoint pairs. · 3. $\vec{BA} = -\vec{u}$; $\vec{AB} + \vec{BA} = \vec{u} + (-\vec{u}) = \vec{0}$. · 4. A vector is defined only by its magnitude and direction, not its position. · 5. $|\vec{QP}| = 7$ cm — the negative reverses direction but keeps the same magnitude.
Q1 (1 mark): $\vec{CD}$ (two-point); $\vec{w}$ (typed); $\mathbf{w}$ (print); $\underline{w}$ (handwriting) [1 — all four correct].
Q2 (2 marks): In parallelogram $WXYZ$, $\vec{WX} = \vec{ZY}$ (opposite sides, same direction $W \to X$ and $Z \to Y$) [1]. $\vec{WX} \neq \vec{XY}$ because although they share point $X$, they point in different directions (one horizontal, one diagonal in a general parallelogram) [1].
Q3 (2 marks): $\vec{BA} = -\vec{AB}$ [1]. The negative of a vector reverses direction but does not change magnitude, so $|\vec{BA}| = |-\vec{AB}| = |\vec{AB}|$. However $\vec{AB} \neq \vec{BA}$ because they point in opposite directions [1].
Five timed questions. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering vector notation questions. Lighter alternative to the boss.
Mark lesson as complete
Tick when you've finished the practice and review.