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: Picture It
Start with something you already know.
Trips on a city grid
- In a city laid out in blocks, a trip can be told as two numbers.
- The first says how many blocks east, the second how many blocks north.
- Take one trip and then another, and the east blocks add up and the north blocks add up.
- Take the same trip three times over, and both numbers triple.
- Take the trip backward, and both numbers flip sign: west instead of east, south instead of north.
A vector is a trip: a list of numbers that is also an arrow from start to finish.
Adding vectors is one trip after another, matching entries added; scaling is the trip repeated or reversed, every entry multiplied.
The picture stops in two places: a scalar like 1.5 is not a whole number of trips, though the rule is the same; and a vector’s length is the straight line from start to finish, not the blocks walked.
One trip of ⟨1, 2⟩, that is 1 block east and 2 blocks north:
| Trip | Where you end |
|---|---|
| ⟨1, 2⟩ three times | ⟨3, 6⟩ |
| ⟨1, 2⟩ backward | ⟨−1, −2⟩ |
| ⟨1, 2⟩, then ⟨4, −1⟩ | ⟨5, 1⟩ |
Each row is entry by entry: nothing from the east number ever leaks into the north number.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
A delivery robot rolls 2 blocks east and 5 blocks north. Then it rolls 6 blocks east and 1 block south. How many blocks north of its start does it end up?
blocks
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: 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 4: 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 5: Watch an Example
One step at a time.
Watch Leo Combine and Measure Two Vectors
u = ⟨4, −2⟩ and v = ⟨3, 1⟩. Leo finds 3u − 2v and then its length.
- Step 1
Scaling comes first. Multiplying u by 3 triples every entry, so 3u = ⟨12, −6⟩: the same direction, three times as long.
Watch Ravi Start, Then Finish It Yourself
u = ⟨−1, 3⟩ and v = ⟨1, 2⟩. Ravi wants 2u − v and its length. He starts; the rest is yours.
- Step 1
Scaling comes first. Doubling every entry of u gives 2u = ⟨−2, 6⟩.
Watch Noor Test Whether Arrows Share a Line
u = ⟨3, −6⟩. Noor asks whether ⟨−1, 2⟩ and ⟨6, −10⟩ lie on the line through u and the origin.
- Step 1
Every multiple of u lies on that line, so the question is whether one single number times u gives the vector. Noor divides matching entries: −1 ÷ 3 = −1/3 and 2 ÷ (−6) = −1/3.
Step 6: Your Turn
Practice makes it stick.
Two Cables on One Hook
Problem 1 of 17
Two cables pull on a hook at the top of a tower. Written as ⟨across, up⟩ in newtons, one pulls ⟨300, −400⟩ and the other pulls ⟨−300, −400⟩. How strong is their total pull, in newtons?
newtons
Next Year’s Price List
Problem 2 of 17
A school store writes its prices as a vector in dollars: ⟨4, 10⟩ for a notebook and a backpack. Next year every price is multiplied by 1.5 and then $2 is added to each, so the new list is 1.5⟨4, 10⟩ + ⟨2, 2⟩. What will a backpack cost, in dollars?
dollars
Add and Stretch
Entry by entry, scaling first, then Pythagoras where a length is asked for.
Problem 3 of 17
What is the second entry of ⟨5, −2⟩ + ⟨−3, 6⟩?
Problem 4 of 17
What is the second entry of −3⟨2, −4⟩?
Problem 5 of 17
u = ⟨2, 3⟩ and v = ⟨−1, 4⟩. What is the first entry of 4u − 3v?
Problem 6 of 17
u = ⟨7, 1⟩ and v = ⟨2, −11⟩. What is the length of u − v?
Problem 7 of 17
What is the first entry of −2(⟨1, −3⟩ + ⟨4, 6⟩)?
Whose Step?
One line on each page is a slip. Find it, then put it right.
Problem 8 of 17
Ana found 3u − 2v for u = ⟨2, −1⟩ and v = ⟨4, 3⟩. One line is a slip. Which?
Ana’s 3u − 2v for u = ⟨2, −1⟩ and v = ⟨4, 3⟩
- 13u = ⟨6, −3⟩
- 22v = ⟨8, 3⟩
- 33u − 2v = ⟨6 − 8, −3 − 3⟩
- Answer= ⟨−2, −6⟩
Problem 9 of 17
Worked correctly, what is the second entry of 3u − 2v for u = ⟨2, −1⟩ and v = ⟨4, 3⟩?
Problem 10 of 17
Dev found u − v and its length for u = ⟨3, −2⟩ and v = ⟨−4, 5⟩. One line is a slip. Which?
Dev’s u − v and its length, for u = ⟨3, −2⟩ and v = ⟨−4, 5⟩
- 1u − v = ⟨3 − (−4), −2 − 5⟩
- 2= ⟨−1, −7⟩
- Answer|u − v| = √(1 + 49) = √50 ≈ 7.1
Problem 11 of 17
Worked correctly, what is the first entry of u − v for u = ⟨3, −2⟩ and v = ⟨−4, 5⟩?
Problem 12 of 17
Maya wanted the length of u + v for u = ⟨3, 4⟩ and v = ⟨6, −4⟩. One line is a slip. Which?
Maya’s length of u + v, for u = ⟨3, 4⟩ and v = ⟨6, −4⟩
- 1|u| = √(9 + 16) = 5
- 2|v| = √(36 + 16) = √52 ≈ 7.2
- Answer|u + v| = 5 + 7.2 = 12.2
Problem 13 of 17
Worked correctly, what is the length of u + v for u = ⟨3, 4⟩ and v = ⟨6, −4⟩?
Pick the Move
Choose the move that gets the answer. You do not need to work either question all the way through.
Problem 14 of 17
u = ⟨17, 3⟩ and v = ⟨14, 8⟩. You want 5u − 5v with the fewest multiplications. Which move?
Problem 15 of 17
You want to know whether ⟨−9, 6⟩ lies on the line through ⟨3, −2⟩ and the origin. Which move decides it?
Same Vector or Same Length?
Judge each one without working every number out. Select every one that fits.
Problem 16 of 17
Select every expression that equals 2(u + v) for every pair of vectors u and v.
Problem 17 of 17
Select every vector that has the same length as ⟨3, −4⟩.
Step 7: 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?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
For which number c does c⟨2, −1⟩ + ⟨1, 5⟩ land on the horizontal axis, with a second entry of 0?
Stretch 2 of 2
A vector w satisfies 3w − ⟨2, 7⟩ = w + ⟨6, −1⟩. What is the second entry of w?
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.
For the grown-up
The misconception this lesson targets is that a vector operation can treat its entries unevenly: scaling only the first entry, subtracting a negative entry as if it were positive, or taking the length of a sum to be the two lengths added. Each leaves a page that looks tidy, because the arithmetic after the slip is honest. The habits that settle them are cheap: every operation touches every entry and never mixes them; a sum is found first and measured after; and a quick sketch of the arrows shows at a glance when a sign has gone astray or a length is too long.