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

Finding your way in three dimensions

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

  1. The vector v = <1, 2, 2> lives in space. What is its magnitude?

    Answer: ______________

  2. 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: ______________

  3. A vector in space is v = <1, 2, 2>. What is its magnitude?

    Answer: ______________

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

    Answer: ______________

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

    Answer: ______________

  6. Write the vector v = <-2, 5, -1> using the standard unit vectors i, j, k.

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

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

    Answer: ______________

  8. Two points in space are A(1, -2, 3) and B(4, 2, 3). What is the distance between A and B?

    Answer: ______________

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

    • <0, 0.6, 0.8>
    • <0, 0.6, minus 0.8>
    • <0, minus 0.6, minus 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 · Squaring, adding, and rooting gives 1 + 4 + 4 = 9, whose root is 3.
  2. 3 · The midpoint is just the average position between the two endpoints, coordinate by coordinate.
  3. 3 · Squaring, adding, and taking the square root gives the length of any vector, no matter how small the numbers look.
  4. 13 · Square each part, add, and root: 9 + 16 + 144 = 169, whose root is 13.
  5. 0 · Average the y pair: (minus 2 + 2) over 2 = 0.
  6. -2i + 5j - k · Each component of a vector becomes the number multiplying the matching unit vector: i for x, j for y, k for z.
  7. 0 · The midpoint's coordinates are just the average of each pair of matching coordinates.
  8. 5 · The distance formula in space is just the magnitude of the vector connecting the two points.
  9. <0, minus 0.6, minus 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