3  Preliminaries - vector properties

Mechanics is all about modelling, meaning that we need to create mathematical models which describe the mechanics in order to solve for interesting entities. The physical problem is first simplified into a mechanical system, using assumptions. We need to keep track of all errors we introduce in such assumptions, this is where experience comes into play. And what we lack in experience we need make up with experimentation.

In this section we will give some definitions of vectors and point out some properties of vectors that will be useful in the following sections.

3.1 Definitions and Notations

We can write \(\overrightarrow{OA}\) to denote the vector from the origin, \(O\) to the point \(A\). This is called an “origin vector” (ortsvektor). We denote such vectors with an arrow on the top \overrightarrow{} . These vectors are also called position vectors and are very useful in mechanics since it is often easy to define a vector from a point to another point. The reason for this has to do with ease of measurement of position in space.

A vector can also be denoted using bold lowercase letters, i.e., \(\mathbb{a}\), more commonly \(\boldsymbol a\) or using an underline \(\underline a\) which is easier to write on the board during a lecture. We will use the notation \(\mathbb{a}\) in this text to denote vectors for consistency with how vectors are denoted in code, this will become clear later.

Another way of denoting position vectors is using the letter \(\mathbb r\), e.g., \(\mathbb r = \overrightarrow{OA}\). This is often used with subscript to denote the direction of the vector between two points, i.e., \(\mathbb r_{AB}\) is the vector from point \(A\) to point \(B\).

Vectors are expressed in matrix form as column vectors, e.g., \(\mathbb a\) can be written on matrix form as \[ \mathbb a = \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix}, \]

where \(a_x\), \(a_y\) and \(a_z\) are the components of the vector in the \(x\), \(y\) and \(z\) directions, respectively. Since matrix notation takes up space in a text like this, we will often write it on the form \(\mathbb a = [a_x, a_y, a_z]^{\mathsf T}\). The superscript \(^\mathsf T\) denotes the transpose of the matrix.

The magnitude (or length) of a vector \(\mathbb{a}\) is denoted by \(|\mathbb{a}|\) or \(a\). We compute the magnitude by using the Pythagorean theorem, which is a special case of the Euclidean norm or 2-norm

\[ a = |\mathbb{a}| := \sqrt{\mathbb{a} \cdot \mathbb{a}} = \sqrt{a_x^2 + a_y^2 + a_z^2}, \]

where \(a_x\), \(a_y\) and \(a_z\) are the Euclidean components of the vector \(\mathbb a\). These are also bases for the vector space \(\mathbb{R}^3\), where the bases can be written as

\[ \mathbb{i}=\begin{bmatrix}1\\ 0\\ 0 \end{bmatrix},\ \mathbb{j}=\begin{bmatrix}0\\ 1\\ 0 \end{bmatrix}\ \text{and }\mathbb{k}=\begin{bmatrix}0\\ 0\\ 1 \end{bmatrix} \]

as such, any vector \(\mathbb{a}\) can be written as a linear combination of these bases \[ \mathbb a = a_x \mathbb{i} + a_y \mathbb{j} + a_z \mathbb{k}. \]

We use almost exclusively the Cartesian coordinate system and learn to model all vectors using it. If we need to express a vector in other bases, we can simply project the vector onto the new bases. Since we are using a computer algebra system, we can easily convert between bases. We can instead really master the Cartesian coordinate system and use it to solve all problems.

Arguably, even more useful is the seperation of vector into a magnitude and a direction. This is done by normalizing the vector, i.e., dividing the vector by its magnitude. The normalized vector is denoted by \(\mathbb e_a\) and is given by

\[ \mathbb a = a \mathbb e_a, \text{ where } \mathbb e_a = \frac{\mathbb a}{|\mathbb a|} = \frac{\mathbb a}{a} \]

A vector can always be reduced to its two components, a magnitude and a direction. We note that the magnitude is the entity that carries the units of the vector, while the direction is a unitless entity. This is the most important property when it comes to modeling in mechanics!

In mechanics, we tend to sidestep from the naming convention of vectors and use capital letters for forces and moments, e.g., \(\mathbb F = F \mathbb e_F\).

We note the distinction between a vector, \(F = [F_x, F_y, F_z]^{\mathsf T}\) which is a columnvector and a point \(\boldsymbol A = (A_x, A_y, A_z)\) which represents a location in space and is not inherently a vector. Since points do not form a vector space, operations like \(\boldsymbol B - \boldsymbol A\) are nonsensicle. Instead, we proper position vectors as the origin point to these points: \(\mathbb r_{AB} = \mathbb r_{OB} - \mathbb r_{OA}\).

3.1.1 A note on angles

In academic settings, we often are tasked with calculating angles between vectors and the bases. In this setting these get special names: directional cosines. There is no practical use for these and no reason in computation other than making hand calculations slightly easier. Using a CAS, we can simply compute the angle between two vectors using the dot product. We will not use directional cosines in this text, but when there is a need to compute angles, we will use the dot product.

One should be extremly skeptical of introducing additional, non-parametric, angles in the modeling process. Even for cases where a force is given using an angle as input one should question the need for this.

Introduction of angles in the modeling process is a common source of errors. The reason for this is that angles are not invariant under coordinate transformations, i.e., if we change the coordinate system, the angle will change. This is not the case for vectors, which are invariant under coordinate transformations. This means that if we have a vector in one coordinate system, we can always find a vector in another coordinate system that has the same magnitude and direction. This is not true for angles, which can be very different in different coordinate systems.

We are often faced with the introduction of degrees of freedom in the modeling process. Here we have a choice to make: Introduce the unknown degrees of freedom as angles or as components in a vector. The latter is the preferred approach, since it is more robust and less sensitive to the choice of definition of the angle.

The trigonometric functions used with these angles might be ill-defined for certain ranges. One should especially avoid using the sine-rule. A more robust approach is to utillize the Pythagorean theorem along with trigonometric functions, which leads to the law of cosines and is much more robust.