A vector is a list of numbers, written as a column, and in two dimensions also an arrow. Add by adding matching entries: place the second arrow at the tip of the first. Scale by multiplying every entry: the arrow stretches, reverses for a negative, collapses for zero. The zero vector adds nothing, and the familiar rules all hold.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Two ways to see it
⟨3, 1⟩ is a list to a programmer and an arrow 3 across and 1 up to a geometer. Linear algebra works because both views describe the same thing, and each catches errors the other hides.
Adding
u = ⟨3, 1⟩, v = ⟨1, 4⟩: u + v = ⟨4, 5⟩. Walk 3 across and 1 up, then 1 across and 4 up: you end 4 across and 5 up. Subtraction is the same with a reversed second arrow: u − v = ⟨2, −3⟩.
Scaling
3u = ⟨9, 3⟩, three times as long, same direction. −v = ⟨−1, −4⟩, the same length pointing the other way. Any multiple of u lies on the line through u and the origin.
The zero vector
⟨0, 0⟩ adds nothing to anything, and 0 times any vector is it. It plays exactly the role 0 plays for numbers, and the origin is where every arrow starts.
The rules are familiar
u + v = v + u; (u + v) + w = u + (v + w); c(u + v) = cu + cv. Check the last: 2(⟨3, 1⟩ + ⟨1, 4⟩) = 2⟨4, 5⟩ = ⟨8, 10⟩, and ⟨6, 2⟩ + ⟨2, 8⟩ = ⟨8, 10⟩. Nothing surprising happens.
Length
The length of ⟨4, 5⟩ is √(16 + 25) = √41 ≈ 6.4, by Pythagoras. It is at most the two lengths added: √10 + √17 ≈ 3.2 + 4.1 = 7.3. The straight path is never longer than the detour.
Step 2: Try It Yourself
Tap and try it out.
u = ⟨3, 1⟩, v = ⟨1, 4⟩. Find u + v, u − v, 3u − 2v, and the length of u + v
- Answeru + v = ⟨3 + 1, 1 + 4⟩ = ⟨4, 5⟩add matching entries
Step 0 of 4
- Vector a(3, 1) · length 3.16
- Vector b(1, 4) · length 4.12
- a + b(4, 5) · length 6.40
The dashed arrow is b again, moved to the tip of a. The sum closes the triangle, and its components are just the x parts added and the y parts added.
Step 3: In Real Life
Cables on a bridge
Each cable pulls with a vector. Adding them gives the total force on the tower; scaling one models a heavier load. Engineers add arrows before they add steel.
Step 4: Watch an Example
One step at a time.
Watch Imani Combine Two Vectors
Imani computes 2u + 3v where u = ⟨1, 4⟩ and v = ⟨2, −1⟩.
- Step 1
She scales u by 2, giving ⟨2, 8⟩, and v by 3, giving ⟨6, −3⟩.
Step 5: Your Turn
Practice makes it stick.
The Two Legs
Problem 1 of 2
A walk of ⟨4, 2⟩ then ⟨3, 5⟩ blocks. What is the first entry of the total displacement?
The Triple
Problem 2 of 2
3 times the vector ⟨2, −5⟩. What is the second entry?
Add and Stretch
1 of 8
⟨2, 7⟩ + ⟨5, 1⟩. What is the first entry?
2 of 8
⟨2, 7⟩ + ⟨5, 1⟩. What is the second entry?
3 of 8
4⟨3, −2⟩. What is the second entry?
4 of 8
0 times any vector. What is every entry of the result?
5 of 8
⟨6, 3⟩ − ⟨2, 3⟩. What is the second entry?
6 of 8
2⟨1, 1⟩ + 3⟨1, 0⟩. What is the first entry?
7 of 8
Match each operation to its geometric effect.
Tap a card on the left to start.
8 of 8
⟨5, 9⟩ + ⟨0, 0⟩. What is the first entry?
Step 6: Quick Check
Show what you know.
Question 1 of 2
3⟨2, 4⟩ + ⟨1, 0⟩. What is the first entry?
Question 2 of 2
What does multiplying a vector by −2 do geometrically?
What You Learned
- A vector is a list of numbers, and in two dimensions it is also an arrow.
- Add matching entries, which places one arrow at the tip of the other; scale every entry, which stretches or reverses it.
- The zero vector adds nothing, and the familiar rules of arithmetic all hold.
- Length is Pythagoras on the entries, and a sum is never longer than its two parts.