Cogito
Multivariable Calculus · Chapter 1 · Lesson 1
Vectors in Three Dimensions
A third coordinate, and everything else stays the same.
12 problems · about 22 minutes · N-VM.A.1, N-VM.B.4, N-VM.B.5
What this lesson teaches
The student adds, scales and finds the magnitude of vectors in two and three dimensions.
- A vector in space has three components and carries direction plus size.
- Vectors add and scale component by component.
- The magnitude of ⟨a, b, c⟩ is the square root of a² + b² + c².
Warm Up
Straightforward practice. Get the method working first.
5 problemsWhat is the magnitude of ⟨3, 4, 12⟩?
Answer 13
Why 13.
What does multiplying a vector by −2 do?
Answer Doubles its length and reverses its direction.
Why Twice as long, pointing the opposite way.
What is the magnitude of ⟨6, 8, 0⟩?
Answer 10
Why A 6-8-10 triangle.
What is the magnitude of ⟨1, 2, 2⟩?
Answer 3
Why 1 + 4 + 4 = 9.
⟨2, 5, 1⟩ + ⟨3, −5, 4⟩. What is the y-component?
Answer 0
Why 5 + (−5).
Build It Up
The same ideas with more to keep track of.
3 problems3⟨2, −1, 4⟩. What is the z-component?
Answer 12
Why Every component triples.
A vector has magnitude 12. What is the magnitude of the unit vector in that direction?
Answer 1
Why That is what "unit" means.
⟨0, 0, 7⟩ points along which axis? Enter 1 for x, 2 for y, 3 for z.
Answer 3
Why Only the third component is non-zero.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each expression by what it produces.
Answer A vector: u + v, 5u · A number: The magnitude of u, The distance between two points
Why A magnitude has no direction left in it.
The distance from (1, 2, 3) to (1, 6, 6). What is it?
Answer 5
Why The differences are 0, 4 and 3.
The Drone: A drone moves ⟨3, 0, 4⟩ metres. How far did it travel from the start, in metres?
Answer 5
Why 5 metres.
The Two Pulls: One rope pulls ⟨4, 1, 0⟩ and another pulls ⟨−1, 2, 0⟩. What is the x-component of the total?
Answer 3
Why 4 + (−1) = 3.