Vectors

A size and a direction

Some quantities are described completely by one number, a size: a temperature of 20°C, a mass of 3 kg, a speed of 20 km/h. These are called scalars.

Others need a direction as well. A wind of 20 km/h from the northwest is not the same as a wind of 20 km/h from the south. Walking 3 km tells you how far someone went, but not where they ended up; walking 3 km north does. A quantity with a size and a direction is called a vector. Distance and speed are scalars; displacement, which is a distance in a stated direction, and velocity, which is a speed in a stated direction, are vectors.

A vector is drawn as an arrow. The length of the arrow is the size of the vector, and the arrowhead shows its direction. A vector is often named with a single letter, such as a. In print the letter is set in bold type, and by hand it is written with a line under it.

Steps across and up

On a grid, a vector is written by its two steps: how far across, then how far up. These two numbers are the components of the vector. The vector a = (3, 2) means 3 across and 2 up. The step across always comes first.

Across means to the right and up means upward. A step to the left or downward has a negative component, so (−4, 1) means 4 to the left and 1 up, and (2, −3) means 2 to the right and 3 down.

The vector a = (3, 2): the arrow runs 3 squares to the right and 2 squares up.

A movement, not a place

A vector records a movement, and the same movement can start anywhere. The arrow from (1, 3) to (4, 5) also runs 3 across and 2 up, so it is the vector a as well. Arrows with the same components are the same vector, wherever they are drawn.

To find the components of an arrow, take its start from its end, one coordinate at a time. The arrow from (2, 5) to (6, 6) has components 6 − 2 = 4 across and 6 − 5 = 1 up, so it is the vector (4, 1).

This is the difference between a point and a vector, even though both can be written as a pair of numbers. The point (3, 2) is one place on the grid. The vector (3, 2) is a move of 3 across and 2 up, which can be made from any place.

Two arrows, one from (0, 0) and one from (1, 3). Each runs 3 across and 2 up, so both are the vector a.

One movement after another

Now make a second movement, b = (2, 3), 2 across and 3 up, starting where a finished. The arrow a ends at (3, 2), so b runs from (3, 2) to (5, 5).

Together the two movements take you from (0, 0) to (5, 5). The single arrow from the start to the finish is the sum, written a + b. It is also called the resultant: one movement that has the same effect as the two.

The components show why the finish is at (5, 5). Across, the two vectors give 3 + 2 = 5; up, they give 2 + 3 = 5. To add vectors, add the steps across together and the steps up together: a + b = (3 + 2, 2 + 3) = (5, 5).

The arrow b = (2, 3) starts where a = (3, 2) finished, at (3, 2), and ends at (5, 5).

The sum a + b runs straight from the start of a to the end of b: 5 across and 5 up.

Here the first arrow is fixed at (3, 1), and the second, (2, 3), starts at its tip. The gold resultant ends at (3 + 2, 1 + 3) = (5, 4). Drag the head of the second arrow anywhere: the resultant’s components are always the two sums.

The length of a vector

The steps across and up of a vector are at right angles, so the arrow is the hypotenuse of a right triangle whose legs are the two components. By Pythagoras, the vector (3, 4) has length $\sqrt{3^2+4^2}=\sqrt{25}=5$√(3² + 4²) = √25 = 5. The length of a vector is also called its magnitude.

The vector a = (3, 2) has length $\sqrt{9+4}=\sqrt{13}$√(9 + 4) = √13, about 3.61, and so does b = (2, 3). Their sum (5, 5) has length $\sqrt{25+25}=\sqrt{50}$√(25 + 25) = √50, about 7.07. That is less than $\sqrt{13}+\sqrt{13}$√13 + √13, about 7.21. The lengths of two vectors add up to the length of their sum only when the two point the same way; otherwise the sum is shorter. Components add; lengths, in general, do not.

The usual mistakes

Reading the step up first. The arrow of a runs 3 across and 2 up, so it is (3, 2), not (2, 3), which is a different arrow.

Counting the grid lines instead of the squares. The components count the squares stepped over, from the start of the arrow to its head.

Adding the lengths of two vectors to get the length of their sum. Add the components, then find the length of the result.

Treating a vector as a place. The vector (3, 2) is a move; it can start at any point.

A boat and a current

A velocity is a vector: a speed in a direction. A boat steering across a river is moved by its engine and by the current at the same time, so its velocity over the ground is the sum of the two velocity vectors. In the application below, the boat’s own velocity is straight across, the current’s is along the river, and their sum, with its length found by Pythagoras, gives the speed the boat really travels at.