8.8Working with Vectors
Define vectors in component form, perform addition and scalar multiplication, find magnitude |v| and direction angle. Compute the dot product u·v = |u||v|cosθ and determine orthogonality.
Vectors are the language of physics and engineering — force, velocity, and acceleration are all vectors. The dot product measures the angle between vectors and is used in work calculations and projections throughout calculus.
Essential Question
How do you represent, add, and scale vectors in component form, and how do you use the dot product to find angles between vectors and determine if they are perpendicular?
Lesson Overview
Component Form
- • v = ⟨a, b⟩ or v = ai + bj
- • Magnitude: |v| = √(a²+b²)
- • Direction: θ = arctan(b/a) (adjust for quadrant)
- • Unit vector: û = v/|v| = ⟨a/|v|, b/|v|⟩
Vector Operations
- • ⟨a,b⟩ + ⟨c,d⟩ = ⟨a+c, b+d⟩
- • k⟨a,b⟩ = ⟨ka, kb⟩
- • ⟨a,b⟩ − ⟨c,d⟩ = ⟨a−c, b−d⟩
Dot Product
- • ⟨a,b⟩ · ⟨c,d⟩ = ac + bd (scalar result)
- • Geometric: u · v = |u||v|cosθ
- • Angle between vectors: cosθ = (u·v) / (|u||v|)
- • Perpendicular (orthogonal): u · v = 0
- • From magnitude & direction: v = |v|⟨cosθ, sinθ⟩
- • Applications: force, velocity, navigation (bearing problems)
Worked Examples
Given u = ⟨3, −4⟩ and v = ⟨−1, 2⟩, find u + v, u − v, 3u, and |u|.
u + v = ⟨3+(−1), −4+2⟩ = ⟨2, −2⟩
u − v = ⟨3−(−1), −4−2⟩ = ⟨4, −6⟩
3u = ⟨3·3, 3·(−4)⟩ = ⟨9, −12⟩
|u| = √(3²+(−4)²) = √(9+16) = √25 = 5
Find the unit vector in the direction of v = ⟨5, −12⟩.
|v| = √(5²+(−12)²) = √(25+144) = √169 = 13
û = v/|v| = ⟨5/13, −12/13⟩
Find the angle between u = ⟨2, 3⟩ and v = ⟨4, −1⟩.
u · v = 2(4) + 3(−1) = 8 − 3 = 5
|u| = √(4+9) = √13
|v| = √(16+1) = √17
cosθ = 5 / (√13 · √17) = 5/√221
θ = arccos(5/√221) ≈ 70.3°
A force of 200 N acts at 35° above horizontal. Write it in component form.
F_x = 200 cos 35° ≈ 200(0.819) ≈ 163.8 N
F_y = 200 sin 35° ≈ 200(0.574) ≈ 114.7 N
F = ⟨163.8, 114.7⟩
Two forces act on an object: F₁ = ⟨50, 30⟩ N and F₂ = ⟨−20, 40⟩ N. Find the resultant force and its magnitude and direction.
R = F₁ + F₂ = ⟨50+(−20), 30+40⟩ = ⟨30, 70⟩
|R| = √(30²+70²) = √(900+4900) = √5800 ≈ 76.2 N
θ = arctan(70/30) ≈ arctan(2.333) ≈ 66.8°
Guided Practice
Given u = ⟨−2, 5⟩ and v = ⟨4, −1⟩, find 2u − 3v and |2u − 3v|.
Hint: Compute 2u and 3v first, then subtract.
Find the unit vector in the direction of v = ⟨−3, 4⟩.
Hint: |v| = √(9+16) = 5.
Find the dot product of u = ⟨3, −2⟩ and v = ⟨4, 6⟩. Are they perpendicular?
Hint: Perpendicular means dot product = 0.
Find the angle between u = ⟨1, 0⟩ and v = ⟨1, 1⟩.
Hint: Use cosθ = (u·v)/(|u||v|).
A plane flies at 300 km/h at bearing N 40° E. Write the velocity vector in component form (x = east, y = north).
Hint: Bearing N 40° E means 40° east of north; the angle from the positive x-axis is 90° − 40° = 50°.
Key Vocabulary
Vector
A quantity with both magnitude and direction, written ⟨a, b⟩ or ai + bj.
Magnitude
|v| = √(a²+b²), the length of the vector.
Direction angle
θ = arctan(b/a) adjusted for quadrant; the angle from the positive x-axis.
Unit vector
A vector of magnitude 1; û = v/|v|.
Dot product
⟨a,b⟩·⟨c,d⟩ = ac + bd; a scalar that encodes the angle between vectors.
Orthogonal
Perpendicular; two vectors are orthogonal if and only if their dot product is 0.
Resultant
The vector sum of two or more vectors; represents the combined effect.
Component form
Expressing a vector as ⟨a, b⟩ where a and b are the horizontal and vertical components.
Check Your Understanding
Interactive Practice — 5 Questions
The magnitude of v = ⟨−5, 12⟩ is:
u = ⟨2, 3⟩, v = ⟨−3, 2⟩. Their dot product is:
The unit vector in the direction of ⟨3, 4⟩ is:
Two vectors are orthogonal when their dot product is:
A vector with magnitude 10 at angle 120° from positive x-axis has components:
Common Mistakes
Computing |u + v| = |u| + |v|
The triangle inequality says |u+v| ≤ |u|+|v|. You must find u+v first, then compute its magnitude.
Forgetting that the dot product is a scalar, not a vector
u·v = ac+bd is a number (scalar). It is NOT a vector. The result has no direction.
Using θ = arctan(b/a) without checking the quadrant
arctan only gives values in (−π/2, π/2). Always check which quadrant ⟨a,b⟩ is in.
Confusing bearing angles with standard angles
Bearing N θ° E means θ° east of north. The standard angle from positive x-axis is 90°−θ.
Math Tips
The dot product formula u·v = |u||v|cosθ is the key to finding angles between vectors. Solve for cosθ = (u·v)/(|u||v|).
Two vectors are perpendicular if and only if their dot product is 0. This is a fast perpendicularity test.
To write a vector from magnitude r and direction θ: v = ⟨r cosθ, r sinθ⟩. This is the same as polar-to-rectangular conversion.
The unit vector û = v/|v| gives the direction of v without the magnitude. Multiplying any scalar by û gives a vector in the same direction.
For navigation problems, set up coordinates with east = +x and north = +y. Convert bearing angles to standard angles before computing components.