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Forward or sideways? Passing as a vector

Every pass has a start and an end, so every pass is a vector. That turns "he never passes forward" from an opinion into a number, and shows why a player's average pass can hide half of what he does.

Beginner Part 6 of Linear Algebra Through Football

Contents

The football question

Every fan has shouted it: "Pass it forward!" Some players seem to go sideways and backwards all afternoon. Others look for the killer ball every time. How would you prove it, one way or the other?

You measure the passes. And a pass, like a run, is a movement from one spot on the pitch to another: a vector.

The concept

Use the same grid as the rest of this series: a 105 m × 68 m pitch, with x running up the pitch towards the opponent's goal and y running across it.

A pass leaves the passer's foot at A = (20, 30) and reaches a team-mate at B = (45, 40). The pass vector is end minus start:

$$\begin{aligned} \mathbf{p} &= B - A \\ &= (45 - 20,\ 40 - 30) \\ &= (25,\ 10) \end{aligned}$$

In plain football

  • 25: the ball went 25 metres up the pitch.
  • 10: and 10 metres across it.
  • A positive first number means a forward pass, a negative one a backward pass. The bigger the second number is compared with the first, the more sideways the pass.

Its length is Pythagoras:

$$\lVert \mathbf{p} \rVert = \sqrt{25^2 + 10^2} = \sqrt{725} \approx 26.9 \text{ m}$$

In plain football

  • The ball travelled about 27 metres in a straight line from passer to receiver.
  • Its angle is about 22° off straight ahead: a forward pass with a little width in it.

Forward isn't the same as closer to goal

Forward passes are a start, but a 25-metre pass along the touchline and a 25-metre pass into the middle aren't worth the same. What a coach usually wants to know is how much closer the ball got to goal.

The centre of the opponent's goal is at (105, 34). Measure the straight-line distance to it before and after the pass:

$$\text{from } A: \sqrt{85^2 + 4^2} \approx 85.1 \text{ m}$$

$$\text{from } B: \sqrt{60^2 + 6^2} \approx 60.3 \text{ m}$$

In plain football

  • Before the pass, the ball was 85.1 m from goal. After it, 60.3 m.
  • So the pass took the ball 24.8 m closer to goal, nearly all of its 25 m forward, because it started and finished close to the middle of the pitch.
  • The same 25 m forward along the touchline, from (20, 5) to (45, 5), gets the ball only 23.2 m closer, because it stays out wide.
The pass is 26.9 m long and 25 m forward, but what counts here is the change in the two gold lines: 85.1 m to goal before it, 60.3 m after.

Data analysts call passes that move the ball a good distance towards goal progressive passes. Providers set their own thresholds for "a good distance", but they're all built on this before-and-after measurement.

Two passers, six passes each

Here are six passes each from two made-up players: a centre-back, Player C, and a central midfielder, Player M. Each pass is a vector in metres.

  • Player C: (5, 12), (3, −15), (20, 2), (−4, 10), (2, −14), (8, −10)
  • Player M: (25, 10), (18, −6), (30, 4), (12, 15), (−5, 3), (22, −8)

Three simple numbers describe them:

Player C Player M
Average pass length 14.4 m 20.8 m
Forward passes 1 of 6 4 of 6
Average pass vector (5.7, −2.5) (17.0, 3.0)

In plain football

  • Average length: add up the lengths and divide by six. M's passes go further.
  • Forward passes here means more forward than sideways: the first number is bigger than the second, ignoring its sign. M goes forward four times out of six, C once.
  • Average pass vector: add the six vectors and divide by six. M's points firmly up the pitch. C's is short and barely forward.

So yes, on these made-up numbers, Player C is the one playing it sideways. But look closely at that average vector.

The average hides the sideways passes

Player C's average pass vector is only about 6.2 m long, yet his average pass is 14.4 m. Where did the rest go?

His passes go left and right: +12, −15, +10, −14. Added together, the sideways parts almost cancel. The average vector says "he passes a few metres forward", when in fact he's playing firm 10- to 15-metre balls across the back line.

Each player's six passes drawn from one point, with up the pitch to the right. The thin arrows are the passes; the gold one is their average. C's passes to either side cancel, leaving a short average arrow. M's mostly point the same way, so his average stays long.

This is a general rule about vectors: averaging them cancels out movements in opposite directions. The average vector tells you a player's overall tendency, not what his passes look like. To see those, you need the average length too, or the average of the size of each part: C's passes average 10.5 m across, whichever way they go.

Show the mathsAverage vectors, and why they're never longer than the average length. Optional.

For passes \(\mathbf{p}_1, \dots, \mathbf{p}_n\), the average pass vector is

$$\bar{\mathbf{p}} = \frac{1}{n} \sum_{i=1}^{n} \mathbf{p}_i$$

and the average pass length is \(\frac{1}{n} \sum_{i} \lVert \mathbf{p}_i \rVert\). By the triangle inequality,

$$\lVert \bar{\mathbf{p}} \rVert \le \frac{1}{n} \sum_{i=1}^{n} \lVert \mathbf{p}_i \rVert$$

with equality only when every pass points the same way. The ratio of the two, 6.2 ÷ 14.4 ≈ 0.43 for Player C and 17.3 ÷ 20.8 ≈ 0.83 for Player M, measures how consistently a player passes in one direction: 1 if every pass goes the same way, near 0 if they point all over the place.

The distance-to-goal change for a pass from \(A\) to \(B\), with the goal at \(G\), is

$$\lVert G - A \rVert - \lVert G - B \rVert$$

which is positive when the pass moves the ball closer to goal.

Why it matters

Once passes are vectors, the arguments in the stands become questions you can answer:

  • Does he pass forward? Count the passes whose first number beats the second.
  • Does he progress the ball? Add up how much closer to goal his passes take it.
  • Where does the team build? Average the pass vectors from each area of the pitch and draw them as arrows: a map of how the ball moves.
  • Who's similar? Put each player's passing numbers in a vector and use distance or cosine similarity to find players who pass alike.

None of that settles whether a sideways pass was the right one. Sometimes it is. But it replaces "he always goes backwards" with evidence. And when the result goes against you, the maths is cheaper than therapy.

Limitations

  • Start and end only. A vector doesn't know whether the pass was on the ground or in the air, or how many defenders it took out.
  • Distance isn't danger. A pass 20 m from goal into a crowded box and one into space can gain the same metres and be worth very different amounts. Expected-threat models try to value where the ball ends up, not just how far it went.
  • Game state shapes passing. A team protecting a lead passes sideways on purpose. Compare players in similar situations.
  • Coordinates differ between providers. As with runs, convert everything to one grid, with the attack always going the same way, before comparing.
  • Six passes is a made-up example. A real profile needs hundreds of passes, across many matches.

Try it yourself

A full-back plays the ball from (30, 4) to (55, 20). Work out the pass vector, its length, and how much closer to goal (105, 34) it takes the ball. Now try (30, 4) to (55, 4), straight down the line. Both go 25 m forward. Which gets the ball closer to goal, and by how much?

Further reading

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