Math 130 Vector Gap Review — Guided Notes
Fill in the blanks as we work through the examples.
1 Component form from points
Rule: vector = terminal point − initial point
Example: from A(−4, 3) to B(1, −2)
v = ⟨, ⟩
Check: this means right and down.
2 Unit vectors
Goal: same direction, length 1.
Example: w = ⟨8, −6⟩
unit = ⟨8/, −6/⟩
unit = ⟨, ⟩
3 Targeted unit-vector model
Example: w = ⟨6, −3⟩
unit = ⟨6/(3√5), −3/(3√5)⟩
unit = ⟨/√5, /√5⟩
decimal ≈ ⟨, ⟩
4 Magnitude + angle → components
Use cosine for horizontal and sine for vertical.
Example: ‖v‖ = 9, θ = 40°
v ≈ ⟨, ⟩
Check the signs using the quadrant.
5 Targeted magnitude-angle model
Example: ‖v‖ = 8, θ = 75°. Round to one decimal place.
v ≈ ⟨8(), 8()⟩
v ≈ ⟨, ⟩
6 Direction angle by quadrant
Find the reference angle first, then adjust.
| Vector signs | Quadrant | Angle |
|---|---|---|
| ⟨+, +⟩ | I | ref |
| ⟨−, +⟩ | II | 180° − ref |
| ⟨−, −⟩ | III | 180° + ref |
| ⟨+, −⟩ | IV | 360° − ref |
7 Targeted Quadrant IV model
Example: v = ⟨12, −9⟩
reference = tan⁻¹(9/12) ≈ °
Quadrant:
θ = 360° − ° = °
8 Fractional scalar multiplication
Multiply every component first.
Example: w = ⟨10,14⟩, u = ⟨−3,4⟩
= ⟨, ⟩ + ⟨−3,4⟩
= ⟨, ⟩
9 Graphical vector addition: count, add, then draw the resultant
Common-tail example: First find each component from the graph. Then add and draw the resultant yourself.
| Arrow | Component |
|---|---|
| A | ⟨, ⟩ |
| B | ⟨, ⟩ |
| C | ⟨, ⟩ |
| r | ⟨, ⟩ |
Then sketch r from the origin.
Scattered-vector example: Even when vectors are separated, count each tail-to-head movement from the graph.
| Arrow | Component |
|---|---|
| A | ⟨, ⟩ |
| B | ⟨, ⟩ |
| C | ⟨, ⟩ |
| r | ⟨, ⟩ |
Then sketch r from the origin.
10 Final practice and checks
Practice: A = ⟨−4,2⟩, B = ⟨7,−1⟩, C = ⟨−2,5⟩
Sketch direction:
Before submitting, check: