Dot Product

Many abstract concepts that make linear algebra a powerful mathematical tool have their roots in plane geometry so we begin our study of dot product by reviewing basic properties of lengths and angles in the real two-dimensional plane ℝ². Guided by these geometrical properties, we formulate properties of dot product that further will be generalized in the following section. Dot product is used to define a magnitude of a vector that is called norm in mathematics.

 

Proof of Pythagorean Theorem: The length of any one side of a plane triangle (it needs not be a right triangle) is determined by the lengths of its other two sides and the cosine of the angle between them. This is a consequence of the Pythagorean theorem. Proof of Law of cosines theorem: The following command finds the length (number of components) of a vector:

The law of cosines implies a familiar fact about plane triangles: the length of one side is not greater than the sum of the lengths of the other two sides.

Proof of Triangle inequality: Let us consider the triangle formed by vectors

b3 = {.75, .75}; a3 = {1.5, .25}; c3 = {b3, a3}; arc3 = {Green, Circle[{0, 0}, .25, {.25, VectorAngle[b3, {.5, 0}]}]}; Show[ Graphics[{arc3, Directive[Red, Thick], Arrow[{{0, 0}, b3}], Directive[Green, Thick], Arrow[{b3, a3}], Directive[Blue, Thick], Arrow[{{0, 0}, a3}]}, PlotRangePadding -> .5, Axes -> True], Epilog -> {Rotate[ Text["\[Theta] = " <> ToString[VectorAngle[a3, b3]/Degree] <> "\[Degree]", {.45, .17}], 15 Degree], Text["b = " <> ToString[b3], {.8, .8}], Text["a= " <> ToString[a3], {1.75, .25}], Text["c", {1.1, .6}]}]

If we place c = 𝑎 − b beginning at b, it forms the third side of a triangle. Inspired by the Pythagorean theorem, we introduce the notation to indicate the Euclidean length of the vector 𝑎. Then The law of cosines tells us that Thus, Hence, and therefore Mathematica has a dedicated command to determine an angle between two vectors by specifying the reference: The left-hand side of Eq.\eqref{EqDot.1} is the dot product of vectors 𝑎 and b (written is rectangular coordinates) and is denoted by The Euclidean length (called norm) and the dot product are related by where we choose a positive branch for the square root (analytic function). Usually, the Euclidean norm is supplied with subscript "2" because there are known many other norms (that are discussed in the following section).

The prime example of dot operation is work that is defined as the scalar product of force and displacement. The presence of cos(θ) ensures the requirement that the work done by a force perpendicular to the displacement is zero.

Example 1: Sailing boat.

Example 2: Sailing boat with current

 

Motivated by two-dimensional case, we extend dot product to a finite-dimensional vector space under arbitrary field. However, this operation between two vectors can be implemented only when a basis is chosen for the corresponding vector space, allowing one to render coordinates .
Using definition of the dot product, we can verify its following properties. For x, y, z ∈ ℂⁿ and k ∈ ℂ, we have

1. x ⋅ x = 0 if and only if x = 0;
2. (x + y) ⋅ z = x ⋅ z + y ⋅ z;
3. x ⋅ (ky) = k(x ⋅ y);
4. \( {\bf x} \cdot {\bf y} = \overline{{\bf y} \cdot {\bf x}} . \)

This list of properties suggests that the dot product it is a positive-definite bilinear mapping where V* is the dual space. However, in complex case, the first component does not enjoy multiplication by a complex scalar, but by its complex conjugate; this means that Note: Dot product for complex vector spaces can be defined as This definition is usually used in mathematics and we mark it as West Coast  . Our definition is mostly used in physics and engineering; correspondingly, we mark it as East Coast. Both definitions lead to the same conclusions and it does not matter which one is applied, but you have to be consistent.

Example 3: Dot product of complex vectors

A unit vector u is a vector whose length equals one: \( {\bf u} \cdot {\bf u} =1 . \) We say that two vectors x and y are perpendicular if their dot product is zero.

One can use the distributive property of the dot product to show that if (a_x, a_y, a_z) and (b_x, b_y, b_z) represent the components of a and b along the axes x, y, and z, then From the definition of the dot product, we can draw an important conclusion. If we divide both sides of a · b = |a| |b| cos θ by |a|, we get Noting that |b| cos θ is simply the projection of b along a, we conclude that in order to find the perpendicular projection of a vector b along another vector a, take dot product of b with \( \hat{\bf e}_a , \) the unit vector along a.

b = {1, 1}; l = {1, .25}; p = a = Projection[b, l]; x = b.a/Norm[a]^2 a; c = b - x; arc1 = {Green, Circle[{0, 0}, .25, {.25, VectorAngle[b, {.5, 0}]}]}; PlanarAngle[{0, 0} -> {b, p}]; Show[ Graphics[{Line[{{p[[1]] + .02, p[[2]] + .07}, {p[[1]] + .03, p[[2]] + .02}}], Line[{{p[[1]] - .01, p[[2]] + .07}, {p[[1]] + .02, p[[2]] + .08}}], arc1, InfiniteLine[{0, 0}, l], Directive[Red, Thick], Arrow[{{0, 0}, b}], Directive[Green, Thick], Arrow[{{0, 0}, c}], Directive[Blue, Thick], Arrow[{{0, 0}, p}], Directive[Black], Directive[Black, Dotted], Line[{c, {1, 1}}], Line[{p, b}]}, PlotRangePadding -> .5, Axes -> True], GridLines -> {{Projection[b, l][[1]]}, {Projection[b, l][[2]]}}, Epilog -> {Rotate[ Text["\[Theta] = " <> ToString[VectorAngle[a, b]/Degree] <> "\[Degree]", {.43, .17}], 15 Degree], Text["b", {1.05, 1.05}], Text["a", {1.2, .25}], Text["c", {-.3, .6}]}]

Example 4: Angle between two vectors

The dot product of any two vectors of the same dimension can be done with the dot operation given as Dot[vector 1, vector 2] or with use of a period “. “ .

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The dot product of two vectors of the same size \( {\bf x} = \left[ x_1 , x_2 , \ldots , x_n \right] \) and \( {\bf y} = \left[ y_1 , y_2 , \ldots , y_n \right] \) (regardless of whether they are columns or rows because Mathematica does not distinguish rows from columns) is the number, denoted either by \( {\bf x} \cdot {\bf y} \) or \( \left\langle {\bf x} , {\bf y} \right\rangle ,\)

when entries are real, or

when entries are complex. Here \( \overline{\bf x} = \overline{a + {\bf j}\, b} = a - {\bf j}\,b \) is a complex conjugate of a complex number x = a + jb.

The dot product of any two vectors of the same dimension can be done with the dot operation given as Dot[vector 1, vector 2] or with use of a period “. “ .

With Euclidean norm ‖·‖₂, the dot product formula defines θ, the angle between two vectors. The dot product was first introduced by the American physicist and mathematician Josiah Willard Gibbs (1839--1903) in the 1880s. ■

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Augustin-Louis Cauchy		Viktor Yakovlevich Bunyakovsky		Hermann Amandus Schwarz

For any norm, the Cauchy--Bunyakovsky--Schwarz (or simply CBS) inequality holds: The inequality for sums was published by the French mathematician and physicist Augustin-Louis Cauchy (1789--1857) in 1821, while the corresponding inequality for integrals was first proved by the Russian mathematician Viktor Yakovlevich Bunyakovsky (1804--1889) in 1859. The modern proof (which is actually a repetition of the Bunyakovsky's one) of the integral inequality was given by the German mathematician Hermann Amandus Schwarz (1843--1921) in 1888. With Euclidean norm, we can define the dot product as where \( \theta \) is the angle between two vectors. ■

 

Subsection: Exercises References: