A vector is a quantity with both magnitude and direction — displacement, velocity, force and acceleration are vectors. A scalar has magnitude only (mass, length, temperature, speed). A vector is drawn as a directed line segment $\overrightarrow{AB}$: $A$ is the initial point (tail), $B$ the terminal point (head), and the arrow fixes the direction. The length $|\overrightarrow{AB}|$ is the magnitude.
Component form, magnitude and the unit vector
In space, fixing the standard unit vectors $\hat{i},\hat{j},\hat{k}$ along the axes, any vector is written in component form $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, where $a_1,a_2,a_3$ are its scalar components. Its magnitude follows from Pythagoras in three dimensions:
A unit vector has magnitude $1$; the unit vector along $\vec{a}$ (its direction, stripped of length) is
To build a vector of any required magnitude $m$ in the direction of $\vec{a}$, scale the unit vector: $m\,\hat{a}=\dfrac{m}{|\vec{a}|}\vec{a}$.
Types of vectors
- Zero (null) vector $\vec{0}$: magnitude $0$; its initial and terminal points coincide, so it has no definite direction. It is the only vector with magnitude $0$.
- Unit vector: magnitude $1$.
- Coinitial vectors: share the same initial point.
- Collinear (parallel) vectors: lie along the same or parallel lines; equivalently $\vec{a}=\lambda\vec{b}$ for some scalar $\lambda$ ($\lambda>0$ same direction, $\lambda<0$ opposite).
- Equal vectors $\vec{a}=\vec{b}$: same magnitude and same direction, regardless of initial point — equivalently equal components.
- Negative of a vector: $-\vec{a}$ has the same magnitude as $\vec{a}$ but opposite direction; $\overrightarrow{BA}=-\overrightarrow{AB}$.
- Position vector: of a point $P(x,y,z)$ relative to the origin $O$, it is $\overrightarrow{OP}=x\hat{i}+y\hat{j}+z\hat{k}$, with $|\overrightarrow{OP}|=\sqrt{x^2+y^2+z^2}$.
Direction cosines and direction ratios
If $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ makes angles $\alpha,\beta,\gamma$ with the positive $x,y,z$ axes, its direction cosines are $l=\cos\alpha,\ m=\cos\beta,\ n=\cos\gamma$, computed as
The components $a_1,a_2,a_3$ themselves (or any numbers proportional to $l,m,n$) are called direction ratios. Direction cosines are unique for a given direction; direction ratios are not (any non-zero multiple works). Notice that the direction cosines are exactly the components of the unit vector $\hat{a}$, so $\hat{a}=l\hat{i}+m\hat{j}+n\hat{k}$.
Vector algebra: addition and scalar multiplication
Vectors add componentwise: $(a_1\hat i+a_2\hat j+a_3\hat k)+(b_1\hat i+b_2\hat j+b_3\hat k)=(a_1+b_1)\hat i+(a_2+b_2)\hat j+(a_3+b_3)\hat k$. Geometrically this is the triangle law (place the tail of $\vec{b}$ at the head of $\vec{a}$; the closing side is $\vec{a}+\vec{b}$) or, equivalently, the parallelogram law (place $\vec{a},\vec{b}$ coinitial; the diagonal is the sum). Addition is commutative and associative, with $\vec{a}+\vec{0}=\vec{a}$ and $\vec{a}+(-\vec{a})=\vec{0}$. Multiplying by a scalar $\lambda$ scales each component: $\lambda\vec{a}=\lambda a_1\hat i+\lambda a_2\hat j+\lambda a_3\hat k$, with $|\lambda\vec{a}|=|\lambda|\,|\vec{a}|$.
Vector joining two points and the section formula
If $A,B$ have position vectors $\vec{a},\vec{b}$, the vector from $A$ to $B$ is the head minus the tail:
The point $R$ dividing $AB$ in the ratio $m:n$ has position vector given by the section formula:
The midpoint is the special internal case $m=n=1$, giving $\vec{r}=\dfrac{\vec{a}+\vec{b}}{2}$.
Deeper Insight — one object, two faithful descriptions. The power of vectors is that the geometric picture (an arrow you can slide around) and the algebraic description (a triple of components) are completely interchangeable. Equal vectors may sit anywhere in space yet share one component triple, which is exactly why a force diagram and a coordinate calculation give the same answer. Most early mistakes come from forgetting this: writing $\overrightarrow{AB}=\vec{a}-\vec{b}$ (tail minus head) instead of $\vec{b}-\vec{a}$, or treating direction ratios as if they were unique. Keep "head minus tail" and "components $=$ magnitude $\times$ direction cosines" in mind and the whole chapter stays consistent.