Cogito
Multivariable Calculus · Chapter 1 · Lesson 2
The Dot Product
A number that measures agreement.
12 problems · about 23 minutes · N-VM.B.4, N-VM.C.11
What this lesson teaches
The student computes dot products and uses them to find angles and to test perpendicularity.
- The dot product multiplies matching components and adds, giving a number.
- u · v = |u||v|cos θ, so it measures the angle between the vectors.
- A dot product of zero means the vectors are perpendicular.
Warm Up
Straightforward practice. Get the method working first.
5 problems⟨2, −1, 3⟩ · ⟨1, 4, 1⟩. What is the result?
Answer 1
Why 1.
A dot product of zero tells you what?
Answer The vectors are perpendicular.
Why They meet at a right angle.
⟨1, 2, 3⟩ · ⟨4, 5, 6⟩. What is the result?
Answer 32
Why 4 + 10 + 18.
⟨1, 0, 0⟩ · ⟨0, 1, 0⟩. What is the result?
Answer 0
Why Two axes at right angles.
u · u where u = ⟨3, 4, 0⟩. What is the result?
Answer 25
Why The magnitude squared.
Build It Up
The same ideas with more to keep track of.
3 problemsTwo vectors of length 5 and 4 meet at 60 degrees. What is the dot product?
Answer 10
Why cos 60 degrees is 0.5.
A dot product is negative. Is the angle greater than 90 degrees? 1 yes, 0 no.
Answer 1
Why Cosine turns negative past a right angle.
⟨5, 0, 0⟩ · ⟨−3, 8, 9⟩. What is the result?
Answer -15
Why Only the first components survive.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each dot product value to what it says about the angle.
Answer Positive → Less than 90 degrees; Zero → Exactly 90 degrees; Negative → More than 90 degrees
Why Follow the sign of the cosine.
u and v both have length 2 and point the same way. What is u · v?
Answer 4
Why cos 0 is 1.
The Sled: A force ⟨6, 0, 0⟩ newtons moves a sled ⟨4, 3, 0⟩ metres. How much work is done, in joules?
Answer 24
Why 6 × 4 + 0 × 3 + 0 × 0 = 24 joules.
The Right Angle: ⟨2, k, 0⟩ is perpendicular to ⟨6, 3, 0⟩. What is k?
Answer -4
Why k = −4.