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Cogito

Multivariable Calculus · Chapter 1 · Lesson 1

Practice Worksheet

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

NameDate

Figure — use these to answer the problems

Move a and b, then read the sum. The resultant is drawn head to tail.
  • Vector a(3, 4) · length 5
  • Vector b(-1, 2) · length 2.24
  • a + b(2, 6) · length 6.32
a, x-component3
a, y-component4
b, x-component-1
b, y-component2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. What is the magnitude of ⟨3, 4, 12⟩?

    Answer

  2. What does multiplying a vector by −2 do?

    • Doubles its length and reverses its direction.
    • Turns it a quarter turn.

  3. What is the magnitude of ⟨6, 8, 0⟩?

    Answer

  4. What is the magnitude of ⟨1, 2, 2⟩?

    Answer

  5. ⟨2, 5, 1⟩ + ⟨3, −5, 4⟩. What is the y-component?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. 3⟨2, −1, 4⟩. What is the z-component?

    Answer

  2. A vector has magnitude 12. What is the magnitude of the unit vector in that direction?

    Answer

  3. ⟨0, 0, 7⟩ points along which axis? Enter 1 for x, 2 for y, 3 for z.

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Sort each expression by what it produces.

    Write each item under the heading it belongs to: u + v · The magnitude of u · 5u · The distance between two points

    A vector

    A number

  2. The distance from (1, 2, 3) to (1, 6, 6). What is it?

    Answer

  3. The Drone

    A drone moves ⟨3, 0, 4⟩ metres. How far did it travel from the start, in metres?

    Answer

  4. The Two Pulls

    One rope pulls ⟨4, 1, 0⟩ and another pulls ⟨−1, 2, 0⟩. What is the x-component of the total?

    Answer