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Two Forms of the Dot Product

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The dot product is an operation for multiplying two vectors to get a scalar value. Consider two vectors a=[a 1 ​ ,…,a N ​ ] and b=[b 1 ​ ,…,b N ​ ].1 Their dot product is denoted a⋅b, and it has two definitions, an algebraic definition and a geometric definition. The algebraic formulation is the sum of the elements after an element-wise multiplication of the two vectors: a⋅b=a 1 ​ b 1 ​ +⋯+a N ​ b N ​ = n=1 ∑ N ​ a n ​ b n ​ . (1) The geometric formulation is the length of a multiplied by the length of b times the cosine of the angle between the two vectors: a⋅b=∥a∥∥b∥cosθ, (2) where ∥v∥ denotes the length (two-norm) of the vector v. The geometric version can be easily visualized (Figure 1) since cosθ= hypotenuse ∥a∥ adjacent ​ ⟹∥a∥cosθ=adjacent. (3) By the geometric definition, the dot product is the multiplication of the length of two vectors after one of the vectors ( a in Figure 1) has been projected onto the other one ( b in Figure 1

Two Forms of the Dot Product --> Home Blog RSS Two Forms of the Dot Product The dot product is often presented as both an algebraic and a geometric operation. The relationship between these two ideas may not be immediately obvious. I prove that they are equivalent and explain why the relationship makes sense. Published 26 June 2018 Two formulations The dot product is an operation for multiplying two vectors to get a scalar value. Consider two vectors a = [ a 1 , … , a N ] \mathbf{a} = [ a_1, \dots, a_N ] a = [ a 1 ​ , … , a N ​ ] and b = [ b 1 , … , b N ] \mathbf{b} = [ b_1, \dots, b_N ] b = [

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