dot product

dot product of two vectors is an operation that takes two equal-length sequences of numbers (vector and ) and returns a single number

• = ith component of vector
• = ith component of vector
• : dimension of the vector space or number of components of either vector

Properties

Aside

A dot product can be written as the product of the transpose vector with the other vector

Distributivity & Associativity

dot product is distributive

and associative

Relationship with 2D vector

(3Blue1Brown) Intuitively, dot product linearly transform one of the vector

Find the magnitude of a vector

The magnitude of a vector is equal to the square root of the dot product of itself

Find the angle between vectors

The angle between two vectors and is computed by their dot product over the product of vector lengths

Find the angle for vector and

Step 1. Compute dot product

Step 2. Compute each vector length

Step 3. Apply formula

Prove orthogonal

Two vectors are orthogonal/perpendicular if and only if their dot product is 0.