The dot product multiplies matching entries and adds them. It is the tool that puts length and angle into linear algebra.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Length
The norm of v is the square root of v dotted with itself. In two dimensions that is the Pythagorean theorem.
Orthogonal
Two vectors are orthogonal when their dot product is zero. In two or three dimensions that is exactly a right angle.
Why the word changes
In ten dimensions you cannot see an angle, but the dot product still works. Orthogonal is the word that survives the loss of the picture.
Orthonormal bases
A basis of mutually orthogonal unit vectors is orthonormal. The standard basis is the obvious example.
Why they are worth having
In an orthonormal basis, coordinates are just dot products. No system needs solving at all.
Right angles in any dimension
Two vectors are orthogonal when their dot product is zero. The definition needs no picture, so it extends to spaces of any dimension where drawing an angle is impossible.
Orthogonal bases are convenient
Coordinates with respect to an orthogonal basis come from dot products directly, with no system to solve. That convenience is why orthogonal bases are sought out wherever possible.
Orthonormal adds unit length
An orthonormal basis is orthogonal with every vector of length 1. Then the matrix of basis vectors has its transpose as its inverse, which makes computation unusually cheap.
Pythagoras generalises
For orthogonal vectors, the squared length of the sum is the sum of the squared lengths. The familiar theorem holds in any dimension, which is one sign the definitions were chosen well.
Step 2: Try It Yourself
Tap and try it out.
- Vector a(3, 1) · length 3.16
- Vector b(-1, 3) · length 3.16
Step 3: Watch an Example
One step at a time.
Watch Idris Test a Right Angle
Idris checks whether ⟨3, 1⟩ and ⟨−1, 3⟩ are orthogonal.
- Step 1
He multiplies the first entries: 3 × (−1) = −3.
Step 4: Your Turn
Practice makes it stick.
The Length
Problem 1 of 2
The norm of ⟨5, 12⟩. What is it?
The Unknown
Problem 2 of 2
⟨4, k⟩ is orthogonal to ⟨2, 4⟩. What is k?
Right Angles
1 of 8
⟨1, 0⟩ · ⟨0, 1⟩. What is the result?
2 of 8
The norm of ⟨6, 8⟩. What is it?
3 of 8
⟨2, 3⟩ · ⟨4, 1⟩. What is the result?
4 of 8
v · v where v = ⟨3, 4⟩. What is the result?
5 of 8
A unit vector. What is its norm?
6 of 8
⟨1, 2, 2⟩. What is its norm?
7 of 8
Sort each pair by whether it is orthogonal.
Tap something to move it.
- Empty
- Empty
8 of 8
The zero vector dotted with anything. What is the result?
Step 5: Quick Check
Show what you know.
Question 1 of 2
The norm of ⟨8, 15⟩. What is it?
Question 2 of 2
What does a dot product of zero mean?
What You Learned
- The dot product brings length and angle into linear algebra.
- A dot product of zero means the vectors are orthogonal, in any number of dimensions.
- An orthonormal basis makes coordinates trivial: each one is just a dot product.