v⃗ · Vector
A number holding size and direction in one — the vector.
'The treasure is 300m from here' — can this map find you the treasure?
Distinguish equal length from equal vectors.
Why a plane must aim to one side of its destination on a windy day.
Open the immersive version →A reading path containing this lesson: Work with several numbers at once · 3/3
Why does this question matter?
Temperature, weight, price — one number suffices. But wind, velocity, force: 'how much' is only half the information. Omit 'which way' and the airplane lands in the wrong city.
A number treating size and direction as one unit was needed. The arrow is the answer — length for size, heading for direction; in coordinates, the component pair (x, y).
Once the rules were in place — adding arrows (chaining moves), scaling them, measuring them (Pythagoras) — physics, games, and AI all began describing the world in this language.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — the treasure map says only '300m from here'. Can you find the treasure?
Reach the treasure with direction commands
From the start 🧭 to the treasure 💎. Each button moves one cell.
Read the idea from another angle
A vector turns 'movement itself' into a number.
Displacement, not position — wherever you start, 'east 3, north 4' is the same arrow. Detaching the arrow from its starting point is the key abstraction.
A game character moves by adding vectors every frame.
position + velocity×time — joystick input, gravity, knockback: all vector addition. A physics engine is an arrow calculator.
Now read the definitions with their conditions
What the treasure hunt and the chaining confirmed, in mathematical language.
Vector notation
Horizontal move x paired with vertical move y — two numbers recording one arrow completely.
Magnitude
The arrow's length — the Pythagorean theorem back on duty. (3,4) has magnitude 5.
Vector addition
Chain the moves and the components add — in pictures, tail of one arrow to tip of the other.
Make one more distinction
The size of the arrow (3, 4) is 3+4=7.
An arrow's length is measured by Pythagoras — √(3²+4²) = 5. Components speak coordinates; magnitude speaks geometry. The bridge between them is √(x²+y²).
Vector addition is just an arithmetic rule.
It IS the act of chaining arrows. Go 'east 2, north 1' then 'east 1, north 3' and you've gone 'east 3, north 4' — component-wise addition merely records this obvious fact.
Where else can this idea help?
Game movement and physics engines
A character's position updates each frame as 'position vector + velocity vector'. A jump's parabola is the velocity vector plus a gravity vector every frame.
Aircraft wind correction
heading vector + wind vector = actual track. Pilots don't point at the destination — they point where the wind cancels.
Composing forces (structures)
Cable tensions in a bridge, loads on a building — structural design is arranging force vectors so they sum to zero.
AI embeddings
Chatbots turn words into vectors of hundreds of dimensions. 'king − man + woman ≈ queen' — arithmetic on meaning, implemented as vector addition.
Explain it in your own words
Try another question
Q1What is the magnitude of (5, 12)?Show answer ▾
√(5²+12²) = √169 = 13. Not the component sum (17) but Pythagoras — an arrow's length is always a hypotenuse.
Q2You walk 4 east, then 4 west. What's the sum of the two moves?Show answer ▾
(4,0) + (−4,0) = (0,0) — the zero vector. You walked 8 (distance) but displaced 0 — 'how far you walked' and 'where you ended up' are different questions.
Q3A character has velocity (2, 3). Doubling its speed gives what vector?Show answer ▾
2×(2,3) = (4,6) — each component doubled (scalar multiplication). Same direction, twice the size. Multiply by a negative and the direction flips.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: Work with several numbers at once · 3/3
Previous concept
Trigonometric Functions
You've already met the arrow (cos θ, sin θ) — now handle arrows in general.
← Trigonometric Functions labLeads to next
Gradient Descent
The compass AI uses to find downhill (the gradient) is a vector — arrows steering learning.
Go to the Gradient Descent lab →Related lab
Pythagorean Theorem
A vector's magnitude √(x²+y²) IS this theorem — the root of the space journey.
Go to the Pythagorean Theorem lab →Related lab
Trigonometric Functions
The unit-circle point (cos θ, sin θ) is a direction vector of length 1.
Go to the Trigonometric Functions lab →