Dot Product - Scalar Product - Canonical/Euclidean/Standard Inner Product (𝑣·𝑢)

the following are equivalent:

• 𝑣·𝑢
• 𝛴_{1≤𝑖≤𝑛}(𝑣_𝑖𝑢_𝑖)
• (length of projected 𝑣 onto 𝑢)·(length of 𝑢)
• (length of projected 𝑢 onto 𝑣)·(length of 𝑣)

for example

• [1, 2, 3]^𝑇·[4, 5, 6]^𝑇 = (1*4) + (2*5) + (3*6) = 32

Intro

Orthogonality

If the dot product of 2 vectors equals 0 then these vectors are orthogonal

see proof: Why Dot Product of 2 Orthogonal Vectors Equals Zero? (Cosine Similarity)

Similarity with Cosine Similarity

see: Dot Product vs Cosine Similarity

Similarity with Linear Transformations of Multi-Dimensional Vectors to 1-Dimensional Vector

the dot product has close ties with the (linear transformation 1×N matrix or projection matrix) that transforms a multi-dimensional vector into a one-dimensional vector (i.e. scalar value)

Consider the 2 Dimensional Case

say the linear transformation 1×2 matrix = [5, 6]

use the matrix to transform the basis vectors 𝑖 and 𝑗 of the 2D vector space:

• [5, 6]𝑖 = [5, 6][1, 0]^𝑇= [5] # basis vector 𝑖 [1, 0] is transformed to [5]
• [5, 6]𝑗 = [5, 6][0, 1]^𝑇= [6] # basis vector 𝑗 [0, 1] is transformed to [6]

therefore any vector (represented in basis vector notation) can be transformed by multiplying its 𝑖 component by 5 and its 𝑗 component by 6. and finally summing the products

Dot Product Induces The Following