# 0.1 Cartesian vectors and tensors: their algebra

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## Topics covered in this chapter

• Definition of a vector
• Examples of vectors
• Scalar multiplication
• Addition of vectors – coplanar vectors
• Unit vectors
• A basis of non-coplanar vectors
• Scalar product – orthogonality
• Directional cosines for coordinate transformation
• Vector product
• Velocity due to rigid body rotations
• Triple scalar product
• Triple vector product
• Second order tensors
• Examples of second order tensors
• Contraction and multiplication
• The vector of an antisymmetric tensor
• Canonical form of a symmetric tensor
Reading Assignment: Chapter 2 of Aris, Appendix A of BSL

The algebra of vectors and tensors will be described here with Cartesian coordinates so the student can see the operations in terms of its components without the complexity of curvilinear coordinate systems.

## Definition of a vector

Suppose ${x}_{i}$ , i.e., ( ${x}_{1}$ , ${x}_{2}$ , ${x}_{3}$ ), are the Cartesian coordinates of a point $P$ in a frame of reference, $0123$ . Let $0\overline{1}\overline{2}\overline{3}$ be another Cartesian frame of reference with the same origin but defined by a rigid rotation. The coordinates of the point $P$ in the new frame of reference is ${\overline{x}}_{j}$ where the coordinates are related to those in the old frame as follows.

$\begin{array}{l}{\overline{x}}_{j}={l}_{ij}\phantom{\rule{5pt}{0ex}}{x}_{i}={l}_{1j}{x}_{1}+{l}_{2j}{x}_{2}+{l}_{3j}{x}_{3}\\ {x}_{i}={l}_{ij}\phantom{\rule{5pt}{0ex}}{\overline{x}}_{j}={l}_{i1}{\overline{x}}_{1}+{l}_{i2}{\overline{x}}_{2}+{l}_{i3}{\overline{x}}_{3}\end{array}$

where l ij are the cosine of the angle between the old and new coordinate systems. Summation over repeated indices is understood when a term or a product appears with a common index.

Cartesian Vector
A Cartesian vector, a, in three dimensions is a quantity with three components ${a}_{1}$ , ${a}_{2}$ , ${a}_{3}$ in the frame of reference 0123, which, under rotation of the coordinate frame to $0\overline{1}\overline{2}\overline{3}$ , become components ${\overline{a}}_{1},{\overline{a}}_{2},{\overline{a}}_{3}$ , where
${\overline{a}}_{j}={l}_{ij}{a}_{i}$

## Examples of vectors

In Cartesian coordinates, the length of the position vector of a point from the origin is equal to the square root of the sum of the square of the coordinates. The magnitude of a vector, a , is defined as follows.

$|\mathbf{a}|={\left({a}_{i}{a}_{i}\right)}^{1/2}$

A vector with a magnitude of unity is called a unit vector . The vector, a /| a |, is a unit vector with the direction of a . Its components are equal to the cosine of the angle between a and the coordinate axis. Some special unit vectors are the unit vectors in the direction of the coordinate axis and the normal vector of a surface.

## Scalar multiplication

If α is a scalar and a is a vector, the product α a is a vector with components, α a i , magnitude α| a |, and the same direction as a .

## Addition of vectors – coplanar vectors

If a and b are vectors with components a i and b i , then the sum of a and b is a vector with components, a i + b i .

The order and association of the addition of vectors are immaterial.

$\begin{array}{l}\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}\\ \left(\mathbf{a}+\mathbf{b}\right)+\mathbf{c}=\mathbf{a}+\left(\mathbf{b}+\mathbf{c}\right)\end{array}$

The subtraction of one vector from another is the same as multiplying one by the scalar (-1) and adding the resulting vectors.

If a and b are two vectors from the same origin, they are colinear or parallel if one is a linear combination of the other, i.e., they both have the same direction. If a and b are two vectors from the same origin, then all linear combination of a and b are in the same plane as a and b , i.,e., they are coplanar . We will prove this statement when we get to the triple scalar product.

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