Vector Inner-Multiplication Operation

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A Vector Inner-Multiplication Operation is a real-valued vector function that is the summation of the vector element multiplications.

  • AKA: Canonical Dot Product, ·, Scalar Product.
  • Context:
    • input: two Vectors.
    • output: a Non-Negative Real Number [0,infinity)
    • It can (typically) be represented as [math]\displaystyle{ x·y }[/math], or as [math]\displaystyle{ ·(x,y) }[/math].
    • It can define a Dot Product Space.
    • It can be used as a Vector Distance Function (such as kernel functions).
    • It can be solved by a Vector Scalar Product.
    • It can have properties of:
      • [math]\displaystyle{ a \cdot b=0 }[/math], for [math]\displaystyle{ a=0 }[/math], or [math]\displaystyle{ b=0 }[/math].
      • [math]\displaystyle{ a \cdot a = |a|^2 }[/math].
      • [math]\displaystyle{ a \cdot b = b \cdot a }[/math].
      • [math]\displaystyle{ (a+b) \cdot c = a \cdot c + b \cdot c }[/math] (Distributive law).
      • [math]\displaystyle{ a \cdot b=|a||b|\cos\theta }[/math], for [math]\displaystyle{ a \neq 0, b \neq 0 }[/math].
      • [math]\displaystyle{ |a \cdot b| \leq |a||b| }[/math] (Schwarz inequality).
      • [math]\displaystyle{ a \cdot b=a_1b_1+a_2b_2+a_3b_3 }[/math], for vectors [math]\displaystyle{ a = a_1i+a_2j+a_3k }[/math] and [math]\displaystyle{ b = b_1i+b_2j+b_3k }[/math].
    • It can be used to calculate the Vector Length Function by [math]\displaystyle{ \sqrt{\cdot(x,y)} }[/math]
    • The real number [math]\displaystyle{ p=|a| \cos \theta }[/math] (the vector component of [math]\displaystyle{ a }[/math]) in the direction of vector [math]\displaystyle{ b }[/math] or the projection of vector [math]\displaystyle{ a }[/math] in the direction of vector [math]\displaystyle{ b }[/math]. So how much the vector [math]\displaystyle{ a }[/math] is projected in the direction of the vector [math]\displaystyle{ b }[/math] can be found out by making dot product of the vector [math]\displaystyle{ a }[/math] with unit vector of [math]\displaystyle{ b (= \frac{b}{|b|}=u) }[/math]. That is [math]\displaystyle{ a \cdot u=a \cdot \frac{b}{|b|}=|a|\frac{|b|}{|b|} \cos \alpha=|a| \cos \alpha= p }[/math].
      • [math]\displaystyle{ |p| }[/math] is the length of the orthogonal projection of [math]\displaystyle{ a }[/math] on a straight line [math]\displaystyle{ l }[/math] in the direction of [math]\displaystyle{ b }[/math].
      • [math]\displaystyle{ p }[/math] may be positive, zero or negative.

  • Example(s):
    • [math]\displaystyle{ (1,1) \cdot (1,1) = 1 \times 1 + 1 \times 1 = 2 }[/math], (notice that the distance is [math]\displaystyle{ \sqrt{2} }[/math]).
    • [math]\displaystyle{ (1,0) \cdot (1,1) = 1 \times 0 + 1 \times 1 = 1 }[/math].
    • [math]\displaystyle{ \cdot((1,0), (0,1)) = 1 \times 0 + 0 \times 1 = 0 }[/math], (two orthogonal vectors).
    • [math]\displaystyle{ \cdot((1,2,3), (6,5,4)) = 28 }[/math].
  • Counter-Example(s):
  • See: Kernel Function, Hyperplane, Cosine Distance Metric, Vector Multiplication, Vector Length Function.



  • (Wikipedia, 2014) ⇒ Retrieved:2014-4-26.
    • In mathematics, the dot product, or scalar product (or sometimes inner product in the context of Euclidean space), is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation can be defined either algebraically or geometrically. Algebraically, it is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the magnitudes of the two vectors and the cosine of the angle between them. The name "dot product" is derived from the centered dot " · " that is often used to designate this operation; the alternative name "scalar product" emphasizes the scalar (rather than vectorial) nature of the result.

      In three-dimensional space, the dot product contrasts with the cross product of two vectors, which produces a pseudovector as the result. The dot product is directly related to the cosine of the angle between two vectors in Euclidean space of any number of dimensions.


    • Let [math]\displaystyle{ u=(u_1,u_2,\ldots,u_n) }[/math] and [math]\displaystyle{ v=(v_1,v_2,\ldots,v_n) }[/math] two vectors on [math]\displaystyle{ k^n }[/math] where [math]\displaystyle{ k }[/math] is a field (like $\mathbb{R}$ or $\mathbb{C}$).

      Then we define the dot product of the two vectors as: [math]\displaystyle{ u\cdot v=u_1v_1+u_2v_2+\cdots+u_nv_n. }[/math] Notice that [math]\displaystyle{ u\cdot v }[/math] is NOT a vector but a scalar (an element from the field $k$).