Vectors in Three Dimensions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Finding your way in three dimensions

Mathematics · Geometry · ages 19-20
Name ______________________   Date ____________

  1. Find the midpoint of the segment connecting P(0, 0, 0) and Q(6, 8, 10). What is the x-coordinate of the midpoint?

    Answer: ______________

  2. What is the vector from P(-1, 2, 3) to Q(2, -2, 9)?

    • <3, -4, 6>
    • <-3, 4, -6>
    • <1, 0, 12>
    • <3, 4, 6>

  3. The vector v = <3, 4, 12> is in space. What is its magnitude?

    Answer: ______________

  4. The standard unit vector j points along the y-axis.

    Circle one:   True   False

  5. What is the y-coordinate of the midpoint of P(1, -2, 3) and Q(4, 2, 3)?

    Answer: ______________

  6. Find the midpoint of P(-4, 1, 5) and Q(6, 7, 9). What is its z-coordinate?

    Answer: ______________

  7. Points A(1, 2, 3) and B(4, 6, 11) are in space. What is the square of the distance between them?

    Answer: ______________

  8. What is the unit vector in the same direction as <8, 15, 0>?

    • <15/17, 8/17, 0>
    • <8, 15, 0>
    • <8/17, 15/17, 0>
    • <8/15, 15/8, 0>

  9. What is the unit vector pointing opposite to <0, 3, 4>?

    • <0, 0.6, 0.8>
    • <0, 0.6, -0.8>
    • <0, -0.6, -0.8>

  10. How do you write v = <-2, 5, -1> with the helpers i, j, k?

    • -2i + 5j - k
    • -2i - 5j + k
    • 2i + 5j - k
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Finding your way in three dimensions W1-mt_o97GclnNC--s1

  1. 3 · The midpoint is just the average position between the two endpoints, coordinate by coordinate.
  2. <3, -4, 6> · Subtract the starting coordinates from the ending coordinates: <2 - (-1), -2 - 2, 9 - 3> = <3, -4, 6>.
  3. 13 · Square each component, add the results, and take the square root: 9 + 16 + 144 = 169, so the magnitude is 13.
  4. True · The helpers i, j, k point one step along x, y, and z in order.
  5. 0 · Average the y-coordinates: (-2 + 2) divided by 2 = 0.
  6. 7 · Average the z-coordinates: (5 + 9) divided by 2 is 14 divided by 2, or 7.
  7. 89 · The coordinate differences are 3, 4, and 8. Their squared sum is 9 + 16 + 64 = 89.
  8. <8/17, 15/17, 0> · The magnitude is 17, so divide each component by 17. The unit vector is <8/17, 15/17, 0>.
  9. <0, -0.6, -0.8> · Normalize first to <0, 0.6, 0.8>, then flip every sign.
  10. -2i + 5j - k · Each part multiplies its helper: x with i, y with j, z with k.
Worksheet · LightMySky