A vector is a list of numbers. Written as a column it becomes an object you can add and stretch, and in two dimensions you can draw it as an arrow.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Two ways to see it
A vector is a list to a programmer and an arrow to a geometer. Linear algebra works because both views describe the same thing.
Adding
Add matching entries. Geometrically, place the second arrow at the tip of the first and draw from the original start to the final tip.
Scaling
Multiplying by a number stretches the arrow. A negative number reverses it, and zero collapses it to a point.
The zero vector
The vector of all zeros adds nothing to anything. It plays the role that 0 plays for numbers.
The rules are familiar
Addition is commutative and associative, and scaling distributes over addition. Nothing surprising happens.
Arrows you can add and stretch
A vector has magnitude and direction and no fixed position. The two allowed operations — addition and scalar multiplication — are what the whole subject is built on, which is why it is called linear.
Both operations are componentwise
Adding vectors adds corresponding entries; scaling multiplies each entry. Geometrically, addition is nose to tail and scaling stretches or reverses. The algebra and the picture agree exactly.
Any number of dimensions
Nothing in the definitions restricts vectors to two or three entries. A vector of a thousand numbers behaves identically, which is what lets linear algebra handle data with many variables.
Vectors need not be arrows
Polynomials and functions can be added and scaled, so they form vector spaces too. Recognising that the subject is about the operations rather than the arrows is the step that makes it general.
Step 2: Try It Yourself
Tap and try it out.
- 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: 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⟩.
Step 4: 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 5: 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.
- Adding places one arrow at the tip of the other; scaling stretches or reverses it.
- The zero vector adds nothing, exactly as 0 does for numbers.