Vector Algebra

Vectors are quantities that have both magnitude and direction, unlike scalars which are just numbers. While a scalar like "5 meters" describes only size, a vector like "5 meters north" describes size and direction. In physics, forces, velocities, and displacements are vectors. In navigation, we combine multiple velocity vectors to find net motion. This chapter develops vector algebra—the rules for combining vectors—treating them as geometric arrows and as algebraic tuples. Understanding vectors is foundational for chapter-11-three-dimensional-geometry, chapter-08-application-of-integrals, and physics applications.

Scalars vs. Vectors

Scalar: A quantity with magnitude only. Examples: temperature (25°C), mass (10 kg), speed (60 km/h)

Vector: A quantity with magnitude and direction. Examples: velocity (60 km/h north), force (100 N upward), displacement (5 m at 30° angle)

Geometric representation: A vector is drawn as an arrow (directed line segment). Length represents magnitude, direction shows the direction.

Algebraic representation: A vector in the plane is written as (a, b) or ai + bj, where (a, b) are components along x and y axes. In 3D: (a, b, c) or ai + bj + ck.

Vector Notation and Components

A vector v from point P to point Q is denoted PQ or v = (v₁, v₂) in 2D, or v = (v₁, v₂, v₃) in 3D.

Magnitude (length): ||v|| = √(v₁² + v₂²) in 2D, or √(v₁² + v₂² + v₃²) in 3D

Unit vector: A vector with magnitude 1. The unit vector in direction of v is:

**û** = **v**/||**v**||

Zero vector: 0 = (0, 0) or (0, 0, 0); magnitude is 0, direction undefined

Vector Operations

Addition and Subtraction

Add component-wise: u + v = (u₁ + v₁, u₂ + v₂)

Geometric interpretation: Place tail of v at head of u. The sum is the vector from origin to final head.

Subtraction: u - v adds u and the negative of v: -v = (-v₁, -v₂)

Scalar Multiplication

Multiply each component: cv = (cv₁, cv₂)

Geometric: Scales magnitude by |c|. If c < 0, reverses direction.

Example: If v = (2, 3) and c = -2, then -2v = (-4, -6), twice the length pointing opposite direction.

Dot Product (Scalar Product)

The dot product u · v combines two vectors into a scalar:

**u** · **v** = u₁v₁ + u₂v₂ (in 2D)
**u** · **v** = u₁v₁ + u₂v₂ + u₃v₃ (in 3D)

Alternative formula: u · v = ||u|| ||v|| cos(θ), where θ is the angle between them

Interpretation:

• u · v > 0: angle is acute (< 90°)
• u · v = 0: vectors are perpendicular (90°)
• u · v < 0: angle is obtuse (> 90°)

Properties:

• u · v = v · u (commutative)
• u · u = ||u||² (dot product with itself is square of magnitude)
• (u + v) · w = u · w + v · w (distributive)

Application: Finding angles between vectors, determining perpendicularity, computing work (force · displacement)

Cross Product (Vector Product) — 3D Only

The cross product u × v produces a vector perpendicular to both u and v:

**u** × **v** = |**i**    **j**    **k**  |
               |u₁   u₂   u₃|
               |v₁   v₂   v₃|

Magnitude: ||u × v|| = ||u|| ||v|| sin(θ), where θ is angle between vectors

Direction: Perpendicular to both, determined by right-hand rule (fingers curl from u toward v, thumb points in direction of u × v)

Properties:

• u × v = -(v × u) (anti-commutative)
• u × u = 0 (cross product with itself is zero)
• If u and v are parallel, u × v = 0

Interpretation: Magnitude equals area of parallelogram with sides u and v. Used for computing areas, torque, and angular momentum.

Scalar Triple Product

The scalar triple product u · (v × w) equals the volume of the parallelepiped with edges u, v, w.

Computed as the determinant:

**u** · (**v** × **w**) = |u₁  u₂  u₃|
                         |v₁  v₂  v₃|
                         |w₁  w₂  w₃|

(Connects to chapter-04-determinants)

Projection of Vectors

The scalar projection of u onto v is:

comp_**v** **u** = (||**u**|| cos(θ)) = (**u** · **v**) / ||**v**||

The vector projection of u onto v is:

proj_**v** **u** = ((**u** · **v**) / ||**v**||²) **v**

Application: Decomposing forces, finding components in specific directions.

Connections to Other Topics

• chapter-11-three-dimensional-geometry: Vectors describe lines and planes in 3D
• chapter-03-matrices: Vectors are columns of matrices; linear transformations via matrices
• chapter-04-determinants: Determinants compute cross products and scalar triple products
• chapter-08-application-of-integrals: Line and surface integrals use vectors
• Physics: Force, velocity, acceleration, torque, angular momentum all vectors

Socratic Questions

1. The dot product u · v = ||u|| ||v|| cos(θ) relates the algebraic definition (component-wise multiplication) to the geometric angle. How does this formula reveal whether vectors point in similar directions versus opposite directions?

1. Why is the cross product u × v defined only in 3D? What would it mean geometrically to compute a cross product in 2D? Can you think of a way to extend the concept?

1. If u and v are perpendicular, explain why u · v = 0 both algebraically and geometrically. How does this property help identify perpendicular vectors?

1. The magnitude ||u × v|| = ||u|| ||v|| sin(θ) represents the area of a parallelogram. How does this compare to the dot product formula with cosine? What's the geometric meaning of each?

1. In the scalar projection of u onto v, we compute (u · v) / ||v||. Why do we divide by ||v|| rather than just taking the dot product? What does dividing by the magnitude accomplish?