Introduction to vectors
Get started with vectors in Year 11 Specialist Mathematics for Queensland (QCAA). A vector carries both size and direction — like displacement, velocity or force — while a scalar such as distance or speed has size only.
You will learn to draw a vector as a directed line segment, find its magnitude and direction, use vector notation and equality, and combine vectors with the triangle rule — the foundation for everything that follows in vectors.
Theory
A vector has both size and direction, unlike a scalar which has size only. This page introduces vectors in Year 11 Specialist Mathematics (QCAA, Queensland): how to draw a vector as a directed line segment, its magnitude and direction, vector notation, equality, scalar multiples, and the triangle rule for combining vectors.
A scalar is a quantity with size only, such as distance, speed or mass. A vector has both a magnitude (size) and a direction. Displacement, velocity and force are vectors: distance is how far you travelled, but displacement also says which way.
A vector is drawn as a directed line segment — an arrow whose length shows the magnitude and whose arrowhead shows the direction. The vector from \(A\) to \(B\) is written \(\overrightarrow{AB}\), or with a bold letter \(\mathbf{a}\), and its magnitude is \(|\mathbf{a}|\). A unit vector has magnitude \(1\).
Two vectors are equal when they have the same magnitude and the same direction — where they start does not matter. A scalar multiple \(k\mathbf{a}\) points the same way as \(\mathbf{a}\) when \(k>0\) and the opposite way when \(k<0\); its length is \(|k|\) times the length of \(\mathbf{a}\). The negative \(-\mathbf{a}\) is the same length as \(\mathbf{a}\) but reversed, and the zero vector \(\mathbf{0}\) has no direction.
Vectors are added with the triangle rule: place them head to tail, and the resultant runs from the tail of the first to the head of the last. Reversing an arrow negates the vector, so a difference \(\mathbf{b}-\mathbf{a}\) can be read as \(-\mathbf{a}\) followed by \(\mathbf{b}\). Any vector can be written as a combination of sums, differences and scalar multiples of others.
For a vector with perpendicular parts of \(v_1\) horizontally and \(v_2\) vertically, the magnitude comes from Pythagoras:
Multiplying a vector by a scalar \(k\) scales its magnitude by \(|k|\):
The triangle rule adds vectors head to tail, and reversing an arrow negates it:
Expressing a vector as a combination
- Read the diagram: note which vectors are given (e.g. \(\overrightarrow{AB}=\mathbf{a}\)) and which one you must find.
- Trace a path from the start point to the end point along arrows you know, going head to tail.
- Reverse where needed: if you must travel against an arrow, use its negative, so \(\overrightarrow{BA}=-\mathbf{a}\).
- Add and simplify: collect the scalar multiples, e.g. \(-\mathbf{a}+\mathbf{b}=\mathbf{b}-\mathbf{a}\); take magnitudes with Pythagoras only at the end.
Displacement is the straight-line vector; the two legs are perpendicular, so use Pythagoras:
| \(|\mathbf{d}|\) | \(=\) | \(\sqrt{5^2+12^2}\) |
| \(=\) | \(\sqrt{25+144}\) | |
| \(=\) | \(\sqrt{169}\) | |
| \(=\) | \(13\) |
The displacement has magnitude \(13\) m.
Take the size of the scalar times the magnitude; the sign only reverses direction:
| \(|-3\mathbf{a}|\) | \(=\) | \(|-3|\,|\mathbf{a}|\) |
| \(=\) | \(3\times 6\) | |
| \(=\) | \(18\) |
Read off the direction from the sign of the scalar:
| \(-3\) | \(<\) | \(0\) |
| \(\Rightarrow\) | \(=\) | \(\text{opposite direction to }\mathbf{a}\) |
\(|-3\mathbf{a}|=18\), pointing the opposite way to \(\mathbf{a}\).
Trace a path from \(A\) to \(B\) through \(O\), reversing the arrow \(OA\):
| \(\overrightarrow{AB}\) | \(=\) | \(\overrightarrow{AO}+\overrightarrow{OB}\) |
| \(=\) | \(-\overrightarrow{OA}+\overrightarrow{OB}\) | |
| \(=\) | \(-\mathbf{a}+\mathbf{b}\) | |
| \(=\) | \(\mathbf{b}-\mathbf{a}\) |
\(\overrightarrow{AB}=\mathbf{b}-\mathbf{a}\).
Add the two displacements head to tail, then collect the scalar multiples:
| \(\overrightarrow{PR}\) | \(=\) | \(\overrightarrow{PQ}+\overrightarrow{QR}\) |
| \(=\) | \(\mathbf{a}+3\mathbf{a}\) | |
| \(=\) | \(4\mathbf{a}\) |
Now take the magnitude, using \(|k\mathbf{a}|=|k|\,|\mathbf{a}|\):
| \(|\overrightarrow{PR}|\) | \(=\) | \(|4\mathbf{a}|\) |
| \(=\) | \(4\times 2\) | |
| \(=\) | \(8\) |
\(\overrightarrow{PR}=4\mathbf{a}\) with magnitude \(8\) m.
Common pitfalls
Frequently asked questions
What is the difference between a scalar and a vector?
A scalar has size only (distance, speed, mass). A vector has both size and direction (displacement, velocity, force).
How do you find the magnitude of a vector?
If its perpendicular parts are \(v_1\) and \(v_2\), the magnitude is \(|\mathbf{v}|=\sqrt{v_1^{\,2}+v_2^{\,2}}\) by Pythagoras.
What does multiplying a vector by a scalar do?
It scales the length by \(|k|\) and keeps the direction if \(k>0\) or reverses it if \(k<0\); so \(|k\mathbf{a}|=|k|\,|\mathbf{a}|\).
When are two vectors equal?
When they have the same magnitude and the same direction. Their starting points do not need to match.
What is the triangle rule for adding vectors?
Place the vectors head to tail; the resultant runs from the tail of the first to the head of the last, so \(\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}\).
How do you write a vector in the opposite direction?
Negate it. \(-\mathbf{a}\) has the same length as \(\mathbf{a}\) but points the opposite way, and \(\overrightarrow{BA}=-\overrightarrow{AB}\).