Vectors
1.0 Introduction
1.0 Introduction
Definition
A vector is a quantity which has both direction and magnitude but scalar is a quantity which has only magnitude.Vector is denoted by $\overrightarrow a $ and read as vector $a$.
The point $A$ from where the vector $\overrightarrow {AB} $ starts is called its initial point, and the point $B$ where it ends is called its terminal point.
The distance between initial and terminal points of a vector is called the magnitude (or length) of the vector, denoted as ${\left| {\overrightarrow a } \right|}$ and read as mod $a$. The arrow ( $ \to $) indicates the direction of the vector. It is always a non-negative scalar or quantity because length can never be negative.
Position Vector
Consider a point $P$ in a space, having coordinates $(x, y, z)$ with respect to the origin $O(0, 0, 0)$. Then, the $\overrightarrow {OP} $ having $O$ and $P$ as its initial and terminal points, respectively, is called the position vector of the point $P$ with respect to $O$. Using distance formula i.e., $$\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2} + {{({z_2} - {z_1})}^2}} $$ The magnitude of $OP$ $(\left| {OP} \right|)$ is given by $$\sqrt {{x^2} + {y^2} + {z^2}} $$
Direction Cosines 
Consider the position vector $\overrightarrow {OP} $ or $\overrightarrow {r} $ of a point $P(x, y, z)$ w.r.t origin as shown in figure. The angles $\alpha ,\beta ,\gamma $ made by the vector $r$ with the positive directions of $x$, $y$ and $z$ axes respectively, are called its direction angles. The cosine values of these angles, i.e., $\cos \alpha $, $\cos \beta $ and $\cos \gamma $ are called direction cosines of the vector $r$ i.e., $\overrightarrow r $, and usually denoted by $l$, $m$ and $n$ respectively.
From the figure, consider the triangle $OAP$ which is right angled, and in this triangle, we have $\cos \alpha = \frac{x}{r}$. Similarly, from the right angled triangles $OBP$ and $OCP$, we may write $\cos \beta = \frac{y}{r},\cos \gamma = \frac{z}{r}$. Thus, the coordinates of the point $P$ may also be expressed as $(lr, mr,nr)$. The numbers $lr$, $mr$ and $nr$ proportional to the direction cosines are called as direction ratios of $\overrightarrow r $, and denoted as $a$, $b$ and $c$, respectively.
Note: ${l^2} + {m^2} + {n^2} = 1$ but ${a^2} + {b^2} + {c^2} \ne 1$, in general.
