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Cogito

Linear Algebra · Chapter 8 · Lesson 1

Practice Worksheet

Orthogonality and the Dot Product

Right angles, in any number of dimensions.

12 problems · about 22 minutes · N-VM.B.4, N-VM.C.11

NameDate

Figure — use these to answer the problems

Turn the two vectors until they meet at a right angle. At that moment the products of the matching entries cancel exactly.
  • Vector a(3, 1) · length 3.16
  • Vector b(-1, 3) · length 3.16
u, first entry3
u, second entry1
v, first entry-1
v, second entry3
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. The norm of ⟨8, 15⟩. What is it?

    Answer

  2. What does a dot product of zero mean?

    • The two vectors are orthogonal.
    • At least one of them is the zero vector.

  3. ⟨1, 0⟩ · ⟨0, 1⟩. What is the result?

    Answer

  4. The norm of ⟨6, 8⟩. What is it?

    Answer

  5. ⟨2, 3⟩ · ⟨4, 1⟩. What is the result?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. v · v where v = ⟨3, 4⟩. What is the result?

    Answer

  2. A unit vector. What is its norm?

    Answer

  3. ⟨1, 2, 2⟩. What is its norm?

    Answer

Stretch Yourself

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

4 problems

  1. Sort each pair by whether it is orthogonal.

    Write each item under the heading it belongs to: ⟨1, 0⟩ and ⟨0, 5⟩ · ⟨2, 1⟩ and ⟨1, −2⟩ · ⟨1, 1⟩ and ⟨1, 1⟩ · ⟨3, 0⟩ and ⟨6, 0⟩

    Orthogonal

    Not orthogonal

  2. The zero vector dotted with anything. What is the result?

    Answer

  3. The Length

    The norm of ⟨5, 12⟩. What is it?

    Answer

  4. The Unknown

    ⟨4, k⟩ is orthogonal to ⟨2, 4⟩. What is k?

    Answer