AP PRECALCULUS — UNIT 4B · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.8A — Vectors
Notes — Magnitude, Direction, and Components
💡 Learning Objectives (4.8.A Part 1)
By the end of this lesson you will be able to:
- Define a vector as a quantity having both magnitude and direction
- Express a vector in component form ⟨a, b⟩ and as an arrow in the plane
- Compute the magnitude and direction angle of a vector
- Find a unit vector in the direction of a given vector
1. What Is a Vector?
A scalar is a quantity with only magnitude — temperature, mass, time. A VECTOR has both MAGNITUDE (a non-negative size) and DIRECTION (where it's pointing). Examples include displacement, velocity, force, and acceleration.
Vectors are usually drawn as ARROWS. The length of the arrow represents the magnitude; the direction the arrow points represents the direction. Two arrows of the same length and direction represent the SAME vector — even if they sit at different points in the plane.
📝 Position is irrelevant
A vector has no fixed location. Two arrows at different starting points but with the same length and direction represent the same vector. This is sometimes called the FREE-VECTOR convention.
2. Component Form
To represent a vector algebraically, we record its horizontal and vertical changes. If a vector goes from point P to point Q, its components are:
v = ⟨q₁ − p₁, q₂ − p₂⟩
The first component is the horizontal change (positive = right, negative = left). The second is the vertical change (positive = up, negative = down). We use angle brackets to distinguish vector components from coordinate points.
📘 Example — Components
- From (1, 2) to (5, 7): v = ⟨5 − 1, 7 − 2⟩ = ⟨4, 5⟩
- From (3, −2) to (−1, 4): v = ⟨−4, 6⟩
- Position vector of (3, −5) (from origin): ⟨3, −5⟩
3. Magnitude
The MAGNITUDE of a vector v = ⟨a, b⟩ is its length:
|v| = √(a² + b²)
This comes straight from the Pythagorean theorem: a vector with horizontal change a and vertical change b is the hypotenuse of a right triangle with legs |a| and |b|.
📘 Example — Compute magnitudes
- |⟨3, 4⟩| = √(9 + 16) = √25 = 5
- |⟨−1, 1⟩| = √(1 + 1) = √2
- |⟨0, −7⟩| = 7
4. Direction Angle
The DIRECTION ANGLE θ of a vector ⟨a, b⟩ is the angle the vector makes with the positive x-axis, measured counter-clockwise:
tan θ = b/a (with quadrant determined by signs of a and b)
⚠️ Watch the quadrant
Just computing arctan(b/a) doesn't always give the right angle. If the vector is in Quadrant II or III, you must add π to the arctan result.
📘 Example — Direction angle
- ⟨3, 3⟩: tan θ = 1, Quadrant I, so θ = π/4 (or 45°)
- ⟨−2, 2⟩: tan θ = −1, Quadrant II, so θ = π − π/4 = 3π/4 (or 135°)
- ⟨0, −4⟩: pointing straight down, θ = 3π/2 (or 270°)
5. From Magnitude and Direction Back to Components
If you know the magnitude |v| and the direction angle θ, you can recover the components:
v = ⟨|v| cos θ, |v| sin θ⟩
📘 Example — Magnitude/direction → components
A vector has magnitude 10 and direction angle 60°.
- Components: ⟨10 cos 60°, 10 sin 60°⟩ = ⟨5, 5√3⟩
- Verify magnitude: √(25 + 75) = √100 = 10 ✓
6. Unit Vectors
A UNIT VECTOR has magnitude 1. The unit vector in the same direction as v = ⟨a, b⟩ is:
û = v / |v| = ⟨a/|v|, b/|v|⟩
Two important standard unit vectors:
- i = ⟨1, 0⟩ — points along positive x-axis
- j = ⟨0, 1⟩ — points along positive y-axis
Any vector can be written in i,j-form: ⟨a, b⟩ = a i + b j.
📘 Example — Unit vector
- v = ⟨6, 8⟩, so |v| = 10
- Unit vector: ⟨6/10, 8/10⟩ = ⟨3/5, 4/5⟩
- Verify: √(9/25 + 16/25) = √1 = 1 ✓
7. Equal Vectors
Two vectors are EQUAL if they have the same components. So u = v iff u₁ = v₁ AND u₂ = v₂. Direction and magnitude alone are not enough — you need to compare components.
8. Summary
- A vector has both magnitude and direction; its position is irrelevant
- Component form ⟨a, b⟩: horizontal change, vertical change
- Magnitude |⟨a, b⟩| = √(a² + b²)
- Direction angle θ: tan θ = b/a (mind the quadrant)
- Components from magnitude and angle: ⟨|v| cos θ, |v| sin θ⟩
- Unit vector: divide each component by the magnitude; standard ones are i and j