Mathematics

Discovering vectors

Three steps right, two steps up: vectors describe movements with direction and length.

What you need first

The coordinate system Pythagoras' theorem

Picture a treasure map: walk 3 steps east and 2 steps north. That instruction is more than a point on the map, it is a movement. A vector describes exactly this kind of shift. It does not tell you where you are, but where you go and how far. With vectors you can describe flight routes, wind directions, or the movement of a game character.

An arrow with direction and length

A vector is drawn as an arrow. It is written down using its components: how many steps in the x direction, how many in the y direction. The vector (3|2) means 3 to the right and 2 up. Negative numbers flip the direction: (−3|0) means 3 steps to the left. Watch the difference from coordinates: the point P(3, 2) is a fixed address in the coordinate system, while the vector (3|2) is a movement. That is why the starting point does not matter for a vector. Whether you begin the shift (3|2) at the origin or in the middle of the map, it is the same vector, because direction and length are the same.

Coordinate grid

P (3 | 2)

Try it: set a point with the two sliders and read off how many steps in the x direction and y direction you need from the origin. Those are exactly the components of the vector from the origin to your point.

Adding vectors

What happens if you carry out two movements one after the other? First (2|1), then (3|2). On paper you place the second arrow so that it starts at the tip of the first. The vector from the start of the first arrow to the tip of the second is the sum. With numbers it is even simpler: you add the components separately. (2|1) and (3|2) become (5|3), because $2 + 3 = 5$ and $1 + 2 = 3$ . Both ways give the same result.

How long is a vector?

\left| \vec{v} \right| = \sqrt{x^2 + y^2}

The bars on the left and right stand for the length of the vector, also called its magnitude. The arrow is the hypotenuse of a right triangle whose legs are the components x and y. That is why the Pythagorean theorem helps here: the vector (3|4) has length

\sqrt{3^2 + 4^2} = \sqrt{25} = 5

Exercises

0 of 6 solved

Time to try it yourself. You can't break anything, every attempt counts.

What does a vector describe?

A vector goes 4 steps to the right and 3 steps up. What is its x component?

In which direction does the vector (0|5) point?

When adding arrows on paper, you place the start of the second arrow at the … of the first arrow.

You add the vectors (2|1) and (3|2). What is the x component of the result?

Match each vector to its direction.

(2|0)

(−4|0)

(0|6)

(0|−3)

Where this leads

The dot product

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