Inner Products, Length and Orthogonality · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Length, angle, and right angles

Mathematics · Linear Algebra · ages 19-20
Name ______________________   Date ____________

  1. Use the inner product <u, v> = u1 v1 + 4 u2 v2. What is the length of v = (3, 2)?

    Answer: ______________

  2. What is the dot product of (2, -1, 3) and (0, 4, 2) in R cubed?

    Answer: ______________

  3. What is the dot product of (2, 3) and (4, minus 1)?

    Answer: ______________

  4. Which vector is orthogonal to (1, 2)?

    • (1, 2)
    • (2, 1)
    • (2, -1)

  5. With orthonormal basis e1 = (3/5, 4/5), e2 = (minus 4/5, 3/5), v = (1, 2) has first coordinate v dot e1. What is it?

    Answer: ______________

  6. The orthogonal basis is u1 = (2, 0), u2 = (0, 3). For v = (6, 9), what is the coefficient of u1 in v = c1 u1 + c2 u2?

    Answer: ______________

  7. What is the largest possible number of mutually orthogonal nonzero vectors in R squared?

    • One nonzero vector
    • Three nonzero vectors
    • Two nonzero vectors
    • Four nonzero vectors

  8. The rule <u, v> = u1 v1 minus u2 v2 is an inner product on R squared. True or false?

    Circle one:   True   False

  9. Fred tests <u, v> = u1 v1 + 2 u2 v2 and finds <(0, 1), (0, 1)> = 2. Does that single check prove it is an inner product?

    • Yes: one positive value proves all the axioms
    • No: all axioms need proof for all vectors
    • No: squared lengths must never exceed one

  10. Use the orthonormal basis e1 = (1, 0), e2 = (0, -1). What is the second coordinate of v = (7, -2) in this basis?

    Answer: ______________

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Answer key

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

Length, angle, and right angles W1-mt_X0HPRGto4W-s1

  1. 5 · Here <v, v> = 9 + 16 = 25, so the length is sqrt(25) = 5.
  2. 2 · Multiply all three matching pairs and add: 0 - 4 + 6 = 2.
  3. 5 · 2 times 4 + 3 times (minus 1) = 8 minus 3 = 5.
  4. (2, -1) · Only (2, -1) gives a zero dot product: 1 times 2 + 2 times (minus 1) = 0.
  5. 2.2 · v dot e1 = 3/5 + 8/5 = 11/5 = 2.2.
  6. 3 · The coefficient is (v dot u1)/(u1 dot u1) = 12/4 = 3.
  7. Two nonzero vectors · Such vectors are independent, so there can be at most two. The coordinate vectors attain that bound.
  8. False · False: positivity fails since <(0, 1), (0, 1)> = minus 1.
  9. No: all axioms need proof for all vectors · One passing test never certifies a rule; every axiom must hold for all vectors.
  10. 2 · The second coordinate is v dot e2 = 7 times 0 + (-2) times (-1) = 2.
Worksheet · LightMySky