Vectors
Senior 2 · Mathematics
Vectors
Lesson 1 of 4
20 min
A moto driver in Kigali is told, "The customer is 3 km away." That is not enough to find the customer. The driver also needs to know which way. Many quantities in mathematics and science only make sense when they come with a direction: a displacement ("2 km towards Kimironko"), a velocity ("60 km/h towards Musanze"), a force ("20 newtons pulling downwards"). A vector is the mathematical tool that carries a size and a direction together.
Vectors are used by pilots and the drones that deliver medical supplies to rural health centres, by engineers who design bridges, and by game programmers who move characters across a screen. In this unit you will draw vectors, write them with numbers, and calculate with them.
A scalar is a quantity that has size (magnitude) only. A vector is a quantity that has both a magnitude and a direction.
| Scalars (size only) | Vectors (size and direction) |
|---|---|
| mass: 50 kg | displacement: 5 km north-east |
| time: 2 hours | velocity: 60 km/h towards Huye |
| distance: 5 km | force: 20 N downwards (a weight) |
| price: 500 Frw | the move "3 right, 4 up" on a grid |
A vector is drawn as a directed line segment: a line segment with an arrowhead. The vector from point A to point B starts at A, its initial point (or tail), and ends at B, its terminal point (or head). It is written AB; in your exercise book, draw a small arrow over the letters AB. A vector can also be named by one small letter, printed in bold as a and underlined (a with a line under it) when handwritten.
Every vector has three characteristics:
BA has the same magnitude and direction as AB but the opposite sense.
On a grid, a vector is described by how far it moves across and how far it moves up. We stack the two numbers in a column vector:
Worked example 1
A is (1, 2) and B is (4, 6). To go from A to B you move 3 units right and 4 units up, so AB = 34.
Figure 1. The vector AB is the move 3 units across and 4 units up. Diagram: Ganzaa.
You do not need to draw the grid every time. Subtract the coordinates of the start from the coordinates of the end:
AB = xB − xAyB − yA
Worked example 2
P is (5, −1) and Q is (2, 3). Then PQ = 2 − 53 − ( − 1) = −34: 3 units left and 4 units up. Going back the other way, QP = 3−4.
Worked example 3
Start at C(−2, 1) and move by the vector 5−3. Add the components to the coordinates: D = ( − 2 + 5, 1 − 3) = (3, −2).
When a column vector has to be written on one line, as in the exercises, we write its components in brackets with the x-component first, for example "the vector (3, 4)".
Tip
End minus start. AB = B − A, component by component. The most common mistake is to work out A − B, which gives BA: every sign comes out wrong.
Always check your answer with a quick sketch: a positive top number must point right, a positive bottom number must point up.
Classify each quantity as a scalar or a vector: (a) a speed of 40 km/h; (b) a velocity of 40 km/h due west; (c) a mass of 3 kg; (d) a weight of 30 N acting downwards.
Solution: (a) scalar, it has no direction; (b) vector; (c) scalar; (d) vector, because a weight is a force with a direction.
A is (2, −3) and B is (7, 1). Find AB.
Solution: AB = 7 − 21 − ( − 3) = 54.
M is (−1, 4) and N is (−6, 0). Find MN and NM.
Solution: MN = −6 − ( − 1)0 − 4 = −5−4. NM is the opposite: 54.
Start at K(3, 5) and move by the vector −42. Where do you finish?
Solution: (3 − 4, 5 + 2) = ( − 1, 7).
Describe the vector 0−6 in words.
Solution: no movement across, 6 units down: a vector pointing straight down with magnitude 6.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
