Vectors and Their Arithmetic

Arrows you can add and stretch.

12 problems · about 22 minutes · N-VM.A.1, N-VM.B.4, N-VM.B.5

What this lesson teaches

The student performs vector addition and scalar multiplication and interprets both geometrically.

A vector is a list of numbers, and in two dimensions it is also an arrow.
Adding places one arrow at the tip of the other; scaling stretches or reverses it.
The zero vector adds nothing, exactly as 0 does for numbers.

Straightforward practice. Get the method working first.

5 problems

3⟨2, 4⟩ + ⟨1, 0⟩. What is the first entry?
Answer 7
Why 7.
What does multiplying a vector by −2 do geometrically?
Answer Doubles the length and reverses the direction.
Why Twice as long, pointing back the other way.
⟨2, 7⟩ + ⟨5, 1⟩. What is the first entry?
Answer 7
Why 2 + 5.
⟨2, 7⟩ + ⟨5, 1⟩. What is the second entry?
Answer 8
Why 7 + 1.
4⟨3, −2⟩. What is the second entry?
Answer -8
Why 4 × (−2).

The same ideas with more to keep track of.

3 problems

0 times any vector. What is every entry of the result?
Answer 0
Why Everything collapses.
⟨6, 3⟩ − ⟨2, 3⟩. What is the second entry?
Answer 0
Why 3 − 3.
2⟨1, 1⟩ + 3⟨1, 0⟩. What is the first entry?
Answer 5
Why 2 + 3.

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

4 problems

Match each operation to its geometric effect.
Answer Adding two vectors → Placing the second arrow at the tip of the first; Multiplying by 3 → Stretching to three times the length; Multiplying by −1 → Turning the arrow to point the opposite way
Why A negative scalar reverses direction.
⟨5, 9⟩ + ⟨0, 0⟩. What is the first entry?
Answer 5
Why The zero vector changes nothing.
The Two Legs: A walk of ⟨4, 2⟩ then ⟨3, 5⟩ blocks. What is the first entry of the total displacement?
Answer 7
Why 7.
The Triple: 3 times the vector ⟨2, −5⟩. What is the second entry?
Answer -15
Why −15.