Engineering Analysis/Vector Basics

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Scalar Product[edit]

A scalar product is a special type of operation that acts on two vectors, and returns a scalar result. Scalar products are denoted as an ordered pair between angle-brackets: <x,y>. A scalar product between vectors must satisfy the following four rules:

  1. \langle x, x\rangle \ge 0, \quad \forall x \in V
  2. \langle x, x\rangle = 0, only if x = 0
  3. \langle x, y\rangle = \langle y, x\rangle
  4. \langle x, cy_1 + dy_2\rangle = c\langle x, y_1\rangle + d\langle x, y_2\rangle

If an operation satisifes all these requirements, then it is a scalar product.

Examples[edit]

One of the most common scalar products is the dot product, that is discussed commonly in Linear Algebra

Norm[edit]

The norm is an important scalar quantity that indicates the magnitude of the vector. Norms of a vector are typically denoted as \|x\|. To be a norm, an operation must satisfy the following four conditions:

  1. \|x\| \ge 0
  2. \|x\| = 0 only if x = 0.
  3. \|cx\| = |c|\|x\|
  4. \|x + y\| \le \|x\| + \|y\|

A vector is called normal if it's norm is 1. A normal vector is sometimes also referred to as a unit vector. Both notations will be used in this book. To make a vector normal, but keep it pointing in the same direction, we can divide the vector by its norm:

\bar{x} = \frac{x}{\|x\|}

Examples[edit]

One of the most common norms is the cartesian norm, that is defined as the square-root of the sum of the squares:

\|x\| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}

Unit Vector[edit]

A vector is said to be a unit vector if the norm of that vector is 1.

Orthogonality[edit]

Two vectors x and y are said to be orthogonal if the scalar product of the two is equal to zero:

\langle x,y\rangle = 0

Two vectors are said to be orthonormal if their scalar product is zero, and both vectors are unit vectors.

Cauchy-Schwartz Inequality[edit]

The cauchy-schwartz inequality is an important result, and relates the norm of a vector to the scalar product:

|\langle x,y\rangle | \leq \|x\|\|y\|

Metric (Distance)[edit]

The distance between two vectors in the vector space V, called the metric of the two vectors, is denoted by d(x, y). A metric operation must satisfy the following four conditions:

  1. d(x,y) \ge 0
  2. d(x,y) = 0 only if x = y
  3. d(x,y) = d(y, x)
  4. d(x,y) \le d(x,z) + d(z, y)

Examples[edit]

A common form of metric is the distance between points a and b in the cartesian plane:

d(a, b)_{{cartesian}} = \sqrt{(x_a - x_b)^2 + (y_a - y_b)^2}