Unit 4 · Topic 4.8

4.8 Vectors

A vector is a quantity with both a size (magnitude) and a direction, written in components as ⟨a, b⟩. You'll find components, magnitudes and unit vectors, add vectors and multiply them by scalars, and use the dot product to find the angle between two vectors.

Key terms

vector
magnitude
component
unit vector
dot product

Vectors and components

A vector is a directed line segment: an arrow. Its starting point is the tail and its ending point is the head. Its length is the magnitude.

The vector from P₁ = (x₁, y₁) to P₂ = (x₂, y₂) has components a = x₂ − x₁ and b = y₂ − y₁, written ⟨a, b⟩. The vector ⟨0, 0⟩ is the zero vector.

A vector's position doesn't matter, only its components. ⟨a, b⟩ points the same way as the segment from the origin to (a, b).

With the unit vectors i = ⟨1, 0⟩ and j = ⟨0, 1⟩, you can also write ⟨a, b⟩ = ai + bj.

Magnitude, direction and unit vectors

Magnitude: ‖⟨a, b⟩‖ = √(a² + b²), from the Pythagorean theorem.

If a vector has magnitude m and makes angle θ with the positive x-axis, its components are ⟨m cos θ, m sin θ⟩, using trig from unit 3.

A unit vector has magnitude 1. To get the unit vector in the direction of a nonzero vector v, multiply v by 1/‖v‖.

Scalar multiplication and addition

Multiplying a vector by a constant k multiplies each component: k⟨a, b⟩ = ⟨ka, kb⟩. The result is parallel to the original; if k < 0 it points the opposite way.

To add vectors, add matching components: ⟨a₁, b₁⟩ + ⟨a₂, b₂⟩ = ⟨a₁ + a₂, b₁ + b₂⟩. Geometrically, place the tail of the second vector at the head of the first; the sum runs from the first tail to the second head.

Vector addition forms a triangle, so the Law of Cosines (c² = a² + b² − 2ab cos C) and the Law of Sines (sin A / a = sin B / b = sin C / c) can find unknown lengths and angles.

The dot product

The dot product multiplies matching components and adds: ⟨a₁, b₁⟩ · ⟨a₂, b₂⟩ = a₁a₂ + b₁b₂. The result is a number, not a vector.

Geometrically, u · v = ‖u‖ ‖v‖ cos θ, where θ is the angle between the vectors. Solving for cos θ gives the angle.

If two nonzero vectors have dot product 0, then cos θ = 0, so θ = π/2: the vectors are perpendicular.

Worked examples

Try each one yourself first, then open the solution.

Example 1
Components, magnitude and unit vector
Find the vector from P(1, −2) to Q(4, 2), its magnitude, and the unit vector in its direction.
Show the solutionHide the solution
1. Step 1: Components: ⟨4 − 1, 2 − (−2)⟩ = ⟨3, 4⟩, or 3i + 4j.
2. Step 2: Magnitude: √(9 + 16) = 5.
3. Step 3: Unit vector: (1/5)⟨3, 4⟩ = ⟨3/5, 4/5⟩. Check: (3/5)² + (4/5)² = 1.
Answer: ⟨3, 4⟩; magnitude 5; unit vector ⟨3/5, 4/5⟩.
Example 2Calculator allowed
The angle between two vectors
Find the angle between u = ⟨2, 1⟩ and v = ⟨−1, 3⟩.
Show the solutionHide the solution
1. Step 1: Dot product: u · v = 2(−1) + 1(3) = 1.
2. Step 2: Magnitudes: ‖u‖ = √5 and ‖v‖ = √10.
3. Step 3: cos θ = 1/(√5 · √10) = 1/√50 ≈ 0.141.
4. Step 4: θ = arccos(1/√50) ≈ 1.429 radians.
Answer: θ ≈ 1.429 radians (about 81.870°).
Example 3Calculator allowed
Trap: the angle in the Law of Cosines
A vector of magnitude 10 points at 120° from the positive x-axis, and a vector ⟨3, 0⟩ is added to it. Find the magnitude of the sum two ways.
Show the solutionHide the solution
1. Step 1: Components: 10⟨cos 120°, sin 120°⟩ = ⟨−5, 5√3⟩. Sum: ⟨−2, 5√3⟩. Magnitude: √(4 + 75) = √79 ≈ 8.888.
2. Step 2: With the Law of Cosines, you need the angle inside the head-to-tail triangle. The two vectors make a 120° angle when tail to tail, so the angle inside the triangle between their sides is 180° − 120° = 60°.
3. Step 3: c² = 10² + 3² − 2(10)(3) cos 60° = 109 − 30 = 79. Same answer.
4. Step 4: Using 120° by mistake gives 109 + 30 = 139, which is wrong.
Answer: √79 ≈ 8.888.

Common mistakes

Subtracting components in the wrong order. The vector from P₁ to P₂ is ⟨x₂ − x₁, y₂ − y₁⟩.
Thinking the dot product is a vector. It's a single number.
Using the tail-to-tail angle in the Law of Cosines instead of the angle inside the triangle.

On the exam

Unit 4 is not on the AP Precalculus Exam, so you'll see this topic on class tests rather than in May.
Expect to compute components, magnitudes, unit vectors and dot products, and to find angles between vectors. Vectors return in AP Calculus BC and AP Physics.