Vector Products

Mathematica has three multiplication commands for vectors: the dot (or inner) and outer products (for arbitrary vectors), and the cross product (for three dimensional vectors).

For three dimensional vectors \( {\bf a} = a_1 \,{\bf i} + a_2 \,{\bf j} + a_3 \,{\bf k} = \left[ a_1 , a_2 , a_3 \right] \) and \( {\bf b} = b_1 \,{\bf i} + b_2 \,{\bf j} + b_3 \,{\bf k} = \left[ b_1 , b_2 , b_3 \right] \) , it is possible to define special multiplication, called the cross-product:

The cross product can be done on two vectors. It is important to note that the cross product is an operation that is only functional in three dimensions. The operation can be computed using the Cross[vector 1, vector 2] operation or by generating a cross product operator between two vectors by pressing [Esc] cross [Esc]. ([Esc] refers to the escape button)

An outer product is the tensor product of two coordinate vectors \( {\bf u} = \left[ u_1 , u_2 , \ldots , u_m \right] \) and \( {\bf v} = \left[ v_1 , v_2 , \ldots , v_n \right] , \) denoted \( {\bf u} \otimes {\bf v} , \) is an m-by-n matrix W of rank 1 such that its coordinates satisfy \( w_{i,j} = u_i v_j . \) The outer product \( {\bf u} \otimes {\bf v} , \) is equivalent to a matrix multiplication \( {\bf u} \, {\bf v}^{\ast} , \) (or \( {\bf u} \, {\bf v}^{\mathrm T} , \) if vectors are real) provided that u is represented as a column \( m \times 1 \) vector, and v as a column \( n \times 1 \) vector. Here \( {\bf v}^{\ast} = \overline{{\bf v}^{\mathrm T}} . \)

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 complex. Here \( \overline{\bf x} = \overline{a + {\bf j}\, b} = a - {\bf j}\,b = {\bf x}^{\ast} {\bf y} \) 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

\[ {\bf x} \cdot {\bf y} = \| {\bf x} \|_2 \, \| {\bf y} \|_2 \, \cos \theta , \]

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. ■

An inner product of two vectors of the same size, usually denoted by \( \left\langle {\bf x} , {\bf y} \right\rangle ,\) is a generalization of the dot product if it satisfies the following properties:

• \( \left\langle {\bf v}+{\bf u} , {\bf w} \right\rangle = \left\langle {\bf v} , {\bf w} \right\rangle + \left\langle {\bf u} , {\bf w} \right\rangle . \)
• \( \left\langle {\bf v} , \alpha {\bf u} \right\rangle = \alpha \left\langle {\bf v} , {\bf u} \right\rangle \) for any scalar α.
• \( \left\langle {\bf v} , {\bf u} \right\rangle = \overline{\left\langle {\bf u} , {\bf v} \right\rangle} , \) where overline means complex conjugate.
• \( \left\langle {\bf v} , {\bf v} \right\rangle \ge 0 , \) and equal if and only if \( {\bf v} = {\bf 0} . \)

The fourth condition in the list above is known as the positive-definite condition. A vector space together with the inner product is called an inner product space. Every inner product space is a metric space. The metric or norm is given by

\[ \| {\bf u} \| = \sqrt{\left\langle {\bf u} , {\bf u} \right\rangle} . \]

The nonzero vectors u and v of the same size are orthogonal (or perpendicular) when their inner product is zero: \( \left\langle {\bf u} , {\bf v} \right\rangle = 0 . \) We abbreviate it as \( {\bf u} \perp {\bf v} . \) If A is an n × n positive definite matrix and u and v are n-vectors, then we can define the weighted Euclidean inner product

\[ \left\langle {\bf u} , {\bf v} \right\rangle = {\bf A} {\bf u} \cdot {\bf v} = {\bf u} \cdot {\bf A}^{\ast} {\bf v} \qquad\mbox{and} \qquad {\bf u} \cdot {\bf A} {\bf v} = {\bf A}^{\ast} {\bf u} \cdot {\bf v} . \]

In particular, if w₁, w₂, ... , w_n are positive real numbers, which are called weights, and if u = ( u₁, u₂, ... , u_n) and v = ( v₁, v₂, ... , v_n) are vectors in ℝⁿ, then the formula

\[ \left\langle {\bf u} , {\bf v} \right\rangle = w_1 u_1 v_1 + w_2 u_2 v_2 + \cdots + w_n u_n v_n \]

defines an inner product on \( \mathbb{R}^n , \) that is called the weighted Euclidean inner product with weights w₁, w₂, ... , w_n. Example 4: Weighted inner product.

Example 4: The Euclidean inner product and the weighted Euclidean inner product (when \( \left\langle {\bf u} , {\bf v} \right\rangle = \sum_{k=1}^n a_k u_k v_k , \) for some positive numbers \( a_k , \ (k=1,2,\ldots , n \) ) are special cases of a general class of inner products on \( \mathbb{R}^n \) called matrix inner product. Let A be an invertible n-by-n matrix. Then the formula

\[ \left\langle {\bf u} , {\bf v} \right\rangle = {\bf A} {\bf u} \cdot {\bf A} {\bf v} = {\bf v}^{\mathrm T} {\bf A}^{\mathrm T} {\bf A} {\bf u} \]

defines an inner product generated by A.

Example 5: Set of integrable functions

Example 5: In the set of integrable functions on an interval [a,b], we can define the inner product of two functions f and g as

\[ \left\langle f , g \right\rangle = \int_a^b \overline{f} (x)\, g(x) \, {\text d}x \qquad\mbox{or} \qquad \left\langle f , g \right\rangle = \int_a^b f(x)\,\overline{g} (x) \, {\text d}x . \]

Then the norm \( \| f \| \) (also called the 2-norm or 𝔏² norm) becomes the square root of

\[ \| f \|^2 = \left\langle f , f \right\rangle = \int_a^b \left\vert f(x) \right\vert^2 \, {\text d}x . \]

In particular, the 2-norm of the function \( f(x) = 5x^2 +2x -1 \) on the interval [0,1] is

\[ \| 2 x^2 +2x -1 \| = \sqrt{\int_0^1 \left( 5x^2 +2x -1 \right)^2 {\text d}x } = \sqrt{7} . \]

Example 6: Set of polynomials

Example 6: Consider a set of polynomials of degree n. If

\[ {\bf p} = p(x) = p_0 + p_1 x + p_2 x^2 + \cdots + p_n x^n \quad\mbox{and} \quad {\bf q} = q(x) = q_0 + q_1 x + q_2 x^2 + \cdots + q_n x^n \]

are two polynomials, and if \( x_0 , x_1 , \ldots , x_n \) are distinct real numbers (called sample points), then the formula

\[ \left\langle {\bf p} , {\bf q} \right\rangle = p(x_0 ) q(x_0 ) + p_1 (x_1 )q(x_1 ) + \cdots + p(x_n ) q(x_n ) \]

defines an inner product, which is called the evaluation inner product at \( x_0 , x_1 , \ldots , x_n . \) ■

The invention of Cartesian coordinates in 1649 by René Descartes (Latinized name: Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.

We can use the dot product to find the angle between two vectors. From the definition of the dot product, we get

\[ {\bf a} \cdot {\bf b} = \langle {\bf a} , {\bf b} \rangle = \| {\bf a} \| \cdot \| {\bf b} \| \,\cos \theta , \]

where θ is the angle between ywo vectors a and b. If the vectors are nonzero, then

\[ \theta = \arccos \left( \frac{{\bf a} \cdot {\bf b}}{\| {\bf a} \| \cdot \| {\bf b} \| } \right) . \]

Example 7: Angle between two vectors

Example 7: What is the angle between i and i + j + 2k?

\begin{align*} \theta &= \arccos \left( \frac{{\bf i} \cdot ({\bf i} + {\bf j} + 2 {\bf k})}{\| {\bf i} \| \cdot \| {\bf i} + {\bf j} + 2 {\bf k} \| } \right) \\ &= \arccos \left( \frac{1}{\sqrt{6}} \right) \approx 1.15026. \end{align*}

Subsection: Exercises

1. What is the angle between the vectors i + j and i + 3j?
2. What is the area of the quadrilateral with vertices at (1, 1), (4, 2), (3, 7) and (2, 3)?

References:

1. Vector addition
2. Tea
3. Milk