Vectors
Vectors: Components, Magnitude and Direction
Quantities with both size and direction, how components turn them into pairs of numbers, and the rules for adding and scaling them.
What this page covers
- Distinguish a vector from a scalar
- Find the components of a vector from two points
- Add and subtract vectors, and multiply by a scalar
- Compute magnitude and direction from components
Vectors and scalars
An object with magnitude and direction. The source's examples are acceleration, velocity and displacement.
A scalar has magnitude only. Mass, temperature and time are scalars; a wind of 20 km/h is a scalar until a direction is attached, at which point it becomes a vector.
Two vectors are equal if they have the same magnitude and the same direction. Where they are drawn does not matter.
The source draws a vector as an arrow from an initial point P to a final point Q, then draws a second identical arrow elsewhere and notes they are the same vector. It also notes that -v ≠ v: reversing the direction produces a different vector.
- Magnitude
- The length of the arrow, written |v|
- Direction
- The angle the arrow makes with a reference axis
- -v
- Same magnitude, opposite direction
- 2v
- Same direction, twice the magnitude
- Zero vector
- 0 = (0, 0); no direction
Addition: the triangle and parallelogram rules
To add u and v, place the tail of v at the head of u. The sum w = u + v runs from the tail of u to the head of v, and is called the resultant.
The source gives both this and the equivalent parallelogram rule, in which the two vectors are drawn from a common point and the sum is the diagonal of the parallelogram they span. The two constructions produce the same arrow.
Subtraction follows: u - v means u + (-v), which is the other diagonal of the same parallelogram. The source draws both diagonals and labels them w - v and v - w, noting they are opposite in direction.
| Law | Statement |
|---|---|
| Commutative | u + v = v + u |
| Associative | u + (v + w) = (u + v) + w |
| Zero vector | v + 0 = v |
| Additive inverse | v + (-v) = 0 |
| Scalar distributive | c(u + v) = cu + cv |
These are exactly the field laws for addition, restated for vectors. That is not a coincidence — the same structure recurs for matrix addition in Week 11.
Components
Working geometrically is slow. Components turn a vector into a pair of numbers and every operation into arithmetic.
The displacement from P(x1, y1) to Q(x2, y2) has components
The source writes a vector as v = <v1, v2> using angle brackets, and notes that the components are vectors in specified directions — the horizontal and vertical parts into which the arrow resolves.
Worked example
For u = (2, 3) and v = (5, -7):
Adding then subtracting should return the original: (7, -4) - (5, -7) = (2, 3) = u ✓.
Magnitude and direction
This is the distance formula. The source derives it exactly that way, writing the length of the displacement from P(x1, y1) to Q(x2, y2) as √(x2 - x1)2 + (y2 - y1)2 before abbreviating to components.
arctan returns an angle between -90° and 90°, so it cannot distinguish (1, 1) from (-1, -1) — both give tan φ = 1. Sketch the vector and adjust by 180° if it points into the second or third quadrant.
Worked example
For v = (5, -7):
The negative angle is measured clockwise from the positive horizontal axis, which is correct for a vector pointing down and to the right.
Resolving into components
The reverse operation — going from magnitude and direction to components — is how a physical quantity given as a bearing and a speed is put into usable form.
Worked example — the source's aircraft
An aircraft flies at 450 km/h on a bearing of 30°. Find its north and east components.
A bearing is measured clockwise from north, so the angle to the north axis is 30° and the angle to the east axis is 60°.
The source uses cos 60° for the east component, which is the same as sin 30° — the angle measured to the other axis. Both give 12.
√389.72 + 2252 = √151866 + 50625 = √202491 ≈ 450 ✓. The components must recombine to the original magnitude.
Bearings are measured clockwise from north, whereas standard mathematical angles are measured anticlockwise from east. Confusing the two conventions swaps the sine and cosine and is the usual source of error in navigation problems.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| Adding magnitudes instead of components | |u + v| ≠ |u| + |v| in general | Equality only when the vectors are parallel |
| Components taken as (x1 - x2, y1 - y2) | Final minus initial | The vector would point backwards |
| Using arctan without a quadrant check | Sketch the vector | (1,1) and (-1,-1) share a tangent |
| Bearings measured from east | Bearings start at north | Clockwise, not anticlockwise |
| Treating -v as equal to v | Direction is reversed | The source states -v ≠ v |
| Forgetting to square before adding in the magnitude | Square each component first | √25 + 49, not 5 + 7 |
Frequently asked questions
What makes something a vector?
Having both magnitude and direction. The source names acceleration, velocity and displacement. A quantity with size alone — mass, temperature, cost — is a scalar.
When are two vectors equal?
When they have the same magnitude and the same direction. Position is irrelevant, so a vector can be moved anywhere in the plane without changing.
How do I find the components from two points?
Subtract the coordinates: the vector from P(x1, y1) to Q(x2, y2) has components (x2 - x1, y2 - y1).
Is the magnitude formula the same as the distance formula?
Yes. |v| = √v12 + v22 is Pythagoras applied to the components, which are exactly the coordinate differences of the endpoints.
Source. Handwritten teaching notes, Week 10, pages 1-3.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.