Dot Product

Definition

Conceptual Overview

Dot product: operation taking two vectors producing a scalar. Measures magnitude of one vector projected onto another. Also called scalar product or inner product in Euclidean space.

Context in Linear Algebra

Essential tool linking algebraic vector representation to geometric properties. Basis for defining length, angle, orthogonality. Fundamental in vector spaces equipped with Euclidean norm.

Notation

Typically denoted as u · v or ⟨u, v⟩. Result is a real number, not a vector.

Algebraic Formula

Componentwise Expression

For vectors u = (u₁, u₂, ..., uₙ), v = (v₁, v₂, ..., vₙ):

u · v = u₁v₁ + u₂v₂ + ... + uₙvₙ

Summation Notation

Compact form using summation:

u · v = ∑i=1n ui vi

Dimensional Requirements

Defined only for vectors of same dimension n. Output is scalar in ℝ.

Geometric Interpretation

Formula with Angle

Dot product relates to lengths and angle θ between vectors:

u · v = ||u|| ||v|| cos θ

Projection Magnitude

Represents length of projection of u onto v scaled by ||v||.

Physical Meaning

Used in physics for work calculation: force · displacement = work done.

Properties

Commutativity

u · v = v · u

Distributivity

u · (v + w) = u · v + u · w

Scalar Multiplication

(c u) · v = c (u · v) = u · (c v), c ∈ ℝ

Positive Definiteness

u · u ≥ 0, equality iff u = 0

Linearity

Linear in each argument separately.

Orthogonality and Angle

Definition of Orthogonality

Vectors u, v orthogonal if u · v = 0.

Angle Computation

Angle θ between vectors found by:

θ = arccos((u · v) / (||u|| ||v||))

Right Angle and Perpendicularity

Orthogonality corresponds to 90° angle, zero projection.

Vector Projections

Projection of u onto v

Vector projection formula:

projv(u) = ((u · v) / (v · v)) v

Scalar Projection

Scalar component of u in direction of v:

compv(u) = (u · v) / ||v||

Applications

Used in decomposing vectors, shadow calculations, orthogonal components.

Applications

Physics

Work = force · displacement. Power, energy calculations.

Computer Graphics

Lighting, shading, angle calculations for rendering.

Machine Learning

Similarity measures, kernel methods using inner products.

Signal Processing

Correlation, projection of signals onto basis functions.

Computational Methods

Algorithmic Steps

Input: vectors u, v of length nInitialize sum = 0For i = 1 to n: sum += u[i] * v[i]Return sum

Computational Complexity

Time complexity: O(n) per dot product. Efficient for sparse vectors.

Numerical Stability

Care with floating-point rounding errors in large dimensions.

Generalizations and Inner Product Spaces

Inner Product Definition

Dot product as special case of inner product in Euclidean space. Inner products satisfy linearity, symmetry, positive-definiteness.

Complex Vector Spaces

Complex inner product uses conjugate symmetry: ⟨u, v⟩ = ∑ u_i conjugate(v_i).

Norm Induction

Norm induced by inner product: ||u|| = sqrt(⟨u, u⟩).

Examples

Example 1: 2D Vectors

u = (3, 4), v = (2, -1)

u · v = 3*2 + 4*(-1) = 6 - 4 = 2

Example 2: Orthogonal Vectors

u = (1, 0), v = (0, 1)

u · v = 1*0 + 0*1 = 0 (orthogonal)

Example 3: Projection

Project u = (3,4) onto v = (1,0):

projv(u) = ((3*1 + 4*0) / (1*1 + 0*0)) (1,0) = 3 (1,0) = (3,0)

Common Mistakes

Confusing Dot and Cross Product

Dot product yields scalar; cross product yields vector orthogonal to originals.

Dimension Mismatch

Cannot compute dot product for vectors of different sizes.

Ignoring Vector Orientation

Sign of dot product depends on angle; negative values indicate obtuse angle.

Assuming Dot Product is a Distance

Dot product is not a metric; describes projection and angle, not distance.

References

• Strang, G., Introduction to Linear Algebra, Wellesley-Cambridge Press, 5th Edition, 2016, pp. 30-65.
• Axler, S., Linear Algebra Done Right, Springer, 3rd Edition, 2015, pp. 45-80.
• Lay, D. C., Linear Algebra and Its Applications, Pearson, 5th Edition, 2015, pp. 50-90.
• Halmos, P. R., Finite-Dimensional Vector Spaces, Springer, 2nd Edition, 1974, pp. 20-60.
• Anton, H., Elementary Linear Algebra, Wiley, 11th Edition, 2012, pp. 35-70.