Dot Product

Mar 25, 2023

The formula for a dot product can be summarised as follows:

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A⋅B = ║A║║B║ cos θ
    = (A₁B₁) + (A₂B₂) # algorithmic

A dot product serves 2 primary uses:

1. Expresses relationship of direction between 2 vectors:
   • 0 is perpendicular.
   • +ve is an acute angle.
   • -ve is an obtuse angle.
2. Yields simplified projections.

Perpendicular Vectors

Perpendicular vectors will result in a 0 value.

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A = [8, 0] 
B = [0, 9]

A⋅B = (8x0) + (0x9)
    = 0

A = [9, 1]
B = [-1, 9]

A⋅B = (9x-1) + (1x9)
    = 0

Acute Angles (<90°)

Will always yield a positive number.

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# 1st quadrant
A = [8, 2]
B = [2, 9]

A⋅B = (8x2) + (2x9)
    = 34

# 2nd quadrant
A = [-8, 2]
B = [-2, 9]

A⋅B = (-8x-2) + (2x9)
    = 34

# 2nd and 3rd quadrant

A = [-8, 2]
B = [-8, -2]

A⋅B = (-8x-8) + (2x-2)
    = 62

Obtuse Angles (>90°)

Will always yield a negative number.

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A = [-10, 1]
B = [1, 9]

A⋅B = (-10x2) + (1x9)
    = -11

nD vectors

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A = [2, 3, 4]
B = [3, 4, 5]

A⋅B = (2x3) + (3x4) + (4x5)
    = 38

Simple Code Sample

The following is a simple code sample using Go generics.

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type Numeric interface {
	~int | ~float32 | ~float64
}

func Dot[T Numeric](a, b []T) (T, error) {
	if len(a) != len(b) {
		return 0, errors.New("unaligned vectors")
	}

	if len(a) == 0 {
		return 0, errors.New("empty vectors")
	}

	var sum T = 0
	for i := range a {
		sum += a[i] * b[i]
	}

	return sum, nil
}

Glossary

cartesian coordinates

A real number coordinate system that uses N pairs of real numbers to specify a points location in the dimensional space relative to the planes intersection.

euclidean space

Ordinary two-, three-, or n-dimensional space represented by positive integers.

matrix

An array of vectors.

scalar

A quantity only possessing magnitude.

tensor

An entity with components that change in a defined way between different coordinate systems.

vector

A quantity possessing both magnitude and direction.

tags: [ machinelearning algebra golang ]