[NOTES/ME-02009]-Vectors as geometrical objects

The vectors are introduced geometrical objects having a magnitude and direction. Then one can define various operations on vectors. These include multiplication by a real number, addition of two vectors, taking dot and cross products of two vectors. A large variety of physical quantities, such as displacement, velocity etc., appear as vectors. The laws of physics are formulated as vector (tensor equation) equations. In order to be able to make numerical predictions and to compare them with experimental data, geometric description of vector physical quantities turns out inadequate,if not useless. While the orbit of a planet around the Sun can be geometrically described as ellipses, but to use laws of physics to make predictions and detailed numerical comparisons observations it is essential to introduce a coordinate system and work with the three components of the position vector. 

Notation & Convention:

We shall use boldface letters, \({\mathbf A,\mathbf B,\mathbf C..}\), to denote vectors. If \(\hat{i},\hat{j},\hat{k}\) are unit vectors along the coordinate axes, a given vector can be expressed as a linear combination of these unit vectors along the three axes. \begin{equation} {\mathbf A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}. \end{equation} Here \(A_x,A_y,A_z\) are the components of vectors in chosen set of axes. We will use the notation \(\vec{A}=(A_x,A_y,A_z )\) to denote the set of the three components of a vector. Also, we shall use \(\widetilde{\sf A}\) to denote the the column vector of components of a vector \({\mathbf A}\). \begin{equation} \widetilde{\sf A} =\begin{pmatrix}A_x\\A_y\\A_z\end{pmatrix}. \end{equation} We will also use notation \(\vec{R} = (x,y,z)\) for the components of vector. Frequently the components will be assembled in a column vector to write \begin{equation} \widetilde{\sf R} = \begin{pmatrix}x\\ y \\ z \end{pmatrix} \end{equation}