Vectors: 2D, 3D, Magnitude, Direction and Position Vectors

A vector is a quantity that has both magnitude (size) and direction. This contrasts with a scalar, which has only magnitude. Vectors are often represented geometrically by an arrow, where the length of the arrow represents the magnitude and the arrowhead indicates the direction. In coordinate form, a 2D vector can be written as a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$ or using unit vectors $\mathbf{i}$ and $\mathbf{j}$ as $x\mathbf{i} + y\mathbf{j}$. A 3D vector is written as $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$ or $x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$, where $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are unit vectors in the direction of the positive x, y, and z axes respectively.

Vectors can be added, subtracted, and multiplied by a scalar. To add or subtract vectors, you add or subtract their corresponding components. For example, if $\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix}$, then $\mathbf{a} + \mathbf{b} = \begin{pmatrix} a_1+b_1 \\ a_2+b_2 \\ a_3+b_3 \end{pmatrix}$ and $\mathbf{a} - \mathbf{b} = \begin{pmatrix} a_1-b_1 \\ a_2-b_2 \\ a_3-b_3 \end{pmatrix}$. To multiply a vector by a scalar $k$, you multiply each component by $k$. So, $k\mathbf{a} = \begin{pmatrix} ka_1 \\ ka_2 \\ ka_3 \end{pmatrix}$.

The magnitude of a vector, also known as its modulus or length, is a scalar quantity representing its size. For a 2D vector $\mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix}$, its magnitude is found using Pythagoras' theorem. For a 3D vector $\mathbf{a} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$, the magnitude extends this concept.

A unit vector is a vector with a magnitude of 1. A unit vector in the same direction as a given vector $\mathbf{a}$ is denoted by $\hat{\mathbf{a}}$ (read as 'a-hat'). It is found by dividing the vector by its magnitude. This process normalises the vector. The unit vector gives the direction of the original vector without regard to its magnitude.

A position vector describes the position of a point in space relative to a fixed origin, O. If A is a point, its position vector is $\vec{OA}$, often denoted by a lowercase bold letter, $\mathbf{a}$. If we have two points A and B with position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, the displacement vector from A to B is given by $\vec{AB} = \mathbf{b} - \mathbf{a}$. This concept is fundamental for solving geometrical problems using vectors, such as determining if points are collinear (lie on the same straight line) or if lines are parallel.