What is a Vector?
While scalars are quantities that have only magnitude (like temperature, mass, or time), vectors are quantities that have both magnitude and direction. Velocity, force, displacement, and magnetic fields are all vector quantities. Understanding vectors is essential for anyone studying physics, engineering, or computer graphics.
A vector is typically represented as a directed line segment—an arrow with a specific length (magnitude) pointing in a specific direction. The starting point is called the tail, and the ending point is called the head.
Vector Notation and Components
Vectors can be written in several forms. In component form, a 2D vector is written as ⟨a, b⟩ or (a, b), where a is the horizontal component and b is the vertical component. In unit vector notation, we use î (horizontal) and ĵ (vertical): v = aî + bĵ.
Example 1: Vector Components
A car travels 50 miles at an angle of 37° north of east. What are its components?
vx = 50 × cos(37°) ≈ 50 × 0.7986 ≈ 39.9 miles east
vy = 50 × sin(37°) ≈ 50 × 0.6018 ≈ 30.1 miles north
The vector can be written as ⟨39.9, 30.1⟩ or 39.9î + 30.1ĵ
Magnitude of a Vector
The magnitude (or length) of a vector v = ⟨a, b⟩ is found using the Pythagorean theorem: |v| = √(a² + b²). For a 3D vector v = ⟨a, b, c⟩, the magnitude is |v| = √(a² + b² + c²).
Example 2: Finding Magnitude
Find the magnitude of v = ⟨3, 4⟩.
|v| = √(3² + 4²) = √(9 + 16) = √25 = 5
This should look familiar—the famous 3-4-5 right triangle!
Vector Operations
Addition: To add vectors, add the corresponding components. v + w = ⟨a + c, b + d⟩. Geometrically, place the tail of w at the head of v; the resultant (sum) goes from the tail of v to the head of w.
Subtraction: v - w = ⟨a - c, b - d⟩. Geometrically, v - w is the same as v + (-w), where -w has the same magnitude as w but opposite direction.
Scalar Multiplication: Multiplying a vector by a scalar k scales its magnitude (but doesn't change its direction unless k is negative, which reverses direction): kv = ⟨ka, kb⟩.
Example 3: Vector Addition
If v = ⟨2, 3⟩ and w = ⟨5, -1⟩, find v + w and 3v.
v + w = ⟨2+5, 3+(-1)⟩ = ⟨7, 2⟩
3v = ⟨3×2, 3×3⟩ = ⟨6, 9⟩
Unit Vectors
A unit vector has magnitude exactly 1. To find the unit vector in the direction of v, divide v by its magnitude: û = v / |v|. The standard unit vectors in 2D are î = ⟨1, 0⟩ (horizontal) and ĵ = ⟨0, 1⟩ (vertical).
Example 4: Finding a Unit Vector
Find the unit vector in the direction of v = ⟨3, 4⟩.
|v| = √(9 + 16) = 5
û = ⟨3/5, 4/5⟩ = ⟨0.6, 0.8⟩
Verification: √(0.6² + 0.8²) = √(0.36 + 0.64) = √1 = 1 ✓
Dot Product
The dot product (also called scalar product) of two vectors multiplies them to produce a scalar. For v = ⟨a, b⟩ and w = ⟨c, d⟩: v · w = ac + bd = |v||w|cos(θ), where θ is the angle between them.
Example 5: Dot Product Calculation
If v = ⟨1, 2⟩ and w = ⟨3, 4⟩, find v · w.
Method 1 (components): v · w = 1×3 + 2×4 = 3 + 8 = 11
Method 2 (magnitudes and angle): Need |v| = √5, |w| = 5, cos(θ) = 11/(√5 × 5)
The dot product is incredibly useful for determining the angle between vectors and for checking perpendicularity. If v · w = 0, the vectors are perpendicular (orthogonal).
Cross Product (3D Vectors)
The cross product of two 3D vectors produces a third vector that is perpendicular to both. For v = ⟨a₁, a₂, a₃⟩ and w = ⟨b₁, b₂, b₃⟩:
v × w = ⟨a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁⟩
The magnitude of the cross product is |v × w| = |v||w|sin(θ), which equals the area of the parallelogram formed by v and w.
Real-World Applications
Physics: Force, velocity, acceleration, and momentum are all vectors. Adding forces acting on an object requires vector addition. The net force determines an object's acceleration.
Engineering: Structural engineers analyze forces in bridges and buildings using vector methods. Every beam experiences forces that must balance for the structure to be stable.
Navigation: Pilots and sailors use vectors to calculate courses. If an airplane's velocity vector and the wind's velocity vector are known, the resulting ground track is their sum.
Computer Graphics: Every movement, rotation, and scaling in video games and CGI is performed using vector mathematics. The direction a character faces, the trajectory of a projectile, and the lighting of a 3D scene all rely on vector operations.