Cogito
Linear Algebra · Chapter 8 · Lesson 1
Orthogonality and the Dot Product
Right angles, in any number of dimensions.
12 problems · about 22 minutes · N-VM.B.4, N-VM.C.11
What this lesson teaches
The student uses the dot product to test orthogonality and to measure length and angle.
- 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.
Warm Up
Straightforward practice. Get the method working first.
5 problemsThe norm of ⟨8, 15⟩. What is it?
Answer 17
Why 17.
What does a dot product of zero mean?
Answer The two vectors are orthogonal.
Why They meet at a right angle.
⟨1, 0⟩ · ⟨0, 1⟩. What is the result?
Answer 0
Why The axes are perpendicular.
The norm of ⟨6, 8⟩. What is it?
Answer 10
Why 36 + 64.
⟨2, 3⟩ · ⟨4, 1⟩. What is the result?
Answer 11
Why 8 + 3.
Build It Up
The same ideas with more to keep track of.
3 problemsv · v where v = ⟨3, 4⟩. What is the result?
Answer 25
Why The norm squared.
A unit vector. What is its norm?
Answer 1
Why That is the definition.
⟨1, 2, 2⟩. What is its norm?
Answer 3
Why 1 + 4 + 4 = 9.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each pair by whether it is orthogonal.
Answer Orthogonal: ⟨1, 0⟩ and ⟨0, 5⟩, ⟨2, 1⟩ and ⟨1, −2⟩ · Not orthogonal: ⟨1, 1⟩ and ⟨1, 1⟩, ⟨3, 0⟩ and ⟨6, 0⟩
Why Compute each dot product and look for zero.
The zero vector dotted with anything. What is the result?
Answer 0
Why Every product is zero.
The Length: The norm of ⟨5, 12⟩. What is it?
Answer 13
Why 13.
The Unknown: ⟨4, k⟩ is orthogonal to ⟨2, 4⟩. What is k?
Answer -2
Why −2.