What does the dot product of two vectors tell you?

The dot product tells you how well two vectors line up, as one number: the two lengths times the cosine of the angle between them. It is positive when they are less than 90° apart, 0 at a right angle and negative beyond that. For ⟨2, 6⟩ and ⟨3, −1⟩ it is 2 × 3 + 6 × (−1) = 0, so they are perpendicular. In the game every part runs from 1 to 9, so each dot product is positive.

How do you convert polar coordinates to rectangular?

To get rectangular coordinates, multiply the length r by cos θ for x and by sin θ for y. Polar coordinates name a point by a distance and an angle; rectangular ones use an across part and an up part. With r = 12 and θ = 30°, y = 12 × sin 30° = 12 × ½ = 6. The game keeps r even and picks angles whose sine or cosine is 0, ½ or 1, sometimes negative, so answers come out whole.

Is there a trick for finding a vector’s magnitude quickly?

The quickest trick is to spot the Pythagorean triple behind the vector: three whole numbers where the squares of the first two add up to the square of the third. The game builds every vector from just five of them, such as 5, 12, 13 and 8, 15, 17, or from a multiple of one. So ⟨16, 12⟩ is ⟨4, 3⟩ taken four times, and its length is 4 × 5 = 20.

Why is my x component negative?

A negative x is the right answer whenever the angle is past 90°, because the arrow then points to the left of where it starts. Since cos 120° = −½, r = 8 and θ = 120° give x = 8 × (−½) = −4, and at 180° x is minus the full length. Type it with the minus key. The y parts here never go negative, since no angle passes 180°.

Is this vector math game free, and how can a teacher use it in class?

Yes, all three vector levels are free in any web browser, and nobody has to register or install a thing. As a class warm-up, ask how many arrows with whole-number parts are exactly 13 long. Pairs hunt for them, such as ⟨5, 12⟩ and ⟨−13, 0⟩, until all 12 turn up; drawn from one point, their tips outline a circle. With the schools plan, the teacher pays per student and sees on a dashboard who is improving.

Math practice · reimagined

Vectors Game

This vector math game gives arrows as pairs like ⟨24, 7⟩ or as a length and an angle, and every answer is whole, from −24 to 162. Get there first, and your fighter strikes while your rival is still squaring and adding.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What is a vector?

A vector is a quantity with a size and a direction, drawn as an arrow. A wind of 20 km/h is only a speed until you say it blows towards the north; then it is a vector. Two arrows are the same vector when they have equal length and point the same way, wherever they start.

Every level of Vectors

Example: the length of ⟨24, 7⟩

The bars around ⟨24, 7⟩ ask for its length: the arrow goes 24 across and 7 up.
Multiply each part by itself: 24 × 24 = 576 and 7 × 7 = 49.
Add the two squares to get 625, then find the number that times itself makes 625: it is 25.
The arrow is 25 long, just over its 24 across, so you type 25.

Learn it step by step

A vector ⟨x, y⟩ is an arrow: go x steps across and y steps up. Its magnitude is just how LONG that arrow is, measured straight from start to tip.

Since the across-step and up-step make a right angle, the arrow is the hypotenuse of a right triangle — so its length is √(x² + y²), the Pythagorean theorem. For ⟨3, 4⟩: √(9 + 16) = √25 = 5. (These are picked so the length always comes out a whole number.)

Square each part, add, take the square root, and type the length.

Tips and common mistakes

Picture θ = 0°: the arrow lies flat on the x-axis, so x is the whole length r. Since cos 0° = 1, x goes with cos, and y with sin.
Adding the two parts is a common mistake: ⟨12, 9⟩ is not 21 long, as the straight arrow cuts the corner. Instead, 144 + 81 = 225, and √225 = 15.
Walk 6 floor tiles, turn a right angle, walk 8: that is vector addition practice, ⟨6, 0⟩ + ⟨0, 8⟩ = ⟨6, 8⟩. A string back to the start measures 10 tiles.

How to play

Vectors reach the screen in three forms: |⟨21, 20⟩| asks for a length, 29; r=10, θ=120°, x=? wants −5, entered with the minus key; ⟨7, 2⟩ · ⟨3, 9⟩ wants a dot product, 39. Type each whole number on the keypad before your rival, and Chilli strikes harder the sooner yours lands.

— Quick answers —

Vectors: the short FAQ

What does the dot product of two vectors tell you?: The dot product tells you how well two vectors line up, as one number: the two lengths times the cosine of the angle between them. It is positive when they are less than 90° apart, 0 at a right angle and negative beyond that. For ⟨2, 6⟩ and ⟨3, −1⟩ it is 2 × 3 + 6 × (−1) = 0, so they are perpendicular. In the game every part runs from 1 to 9, so each dot product is positive.
How do you convert polar coordinates to rectangular?: To get rectangular coordinates, multiply the length r by cos θ for x and by sin θ for y. Polar coordinates name a point by a distance and an angle; rectangular ones use an across part and an up part. With r = 12 and θ = 30°, y = 12 × sin 30° = 12 × ½ = 6. The game keeps r even and picks angles whose sine or cosine is 0, ½ or 1, sometimes negative, so answers come out whole.
Is there a trick for finding a vector’s magnitude quickly?: The quickest trick is to spot the Pythagorean triple behind the vector: three whole numbers where the squares of the first two add up to the square of the third. The game builds every vector from just five of them, such as 5, 12, 13 and 8, 15, 17, or from a multiple of one. So ⟨16, 12⟩ is ⟨4, 3⟩ taken four times, and its length is 4 × 5 = 20.
Why is my x component negative?: A negative x is the right answer whenever the angle is past 90°, because the arrow then points to the left of where it starts. Since cos 120° = −½, r = 8 and θ = 120° give x = 8 × (−½) = −4, and at 180° x is minus the full length. Type it with the minus key. The y parts here never go negative, since no angle passes 180°.
Is this vector math game free, and how can a teacher use it in class?: Yes, all three vector levels are free in any web browser, and nobody has to register or install a thing. As a class warm-up, ask how many arrows with whole-number parts are exactly 13 long. Pairs hunt for them, such as ⟨5, 12⟩ and ⟨−13, 0⟩, until all 12 turn up; drawn from one point, their tips outline a circle. With the schools plan, the teacher pays per student and sees on a dashboard who is improving.

More math to master

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