A vector has both magnitude and direction (e.g. displacement, force), unlike a scalar which has magnitude only. In space a vector is written $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$.
Magnitude and unit vectors
The magnitude is $|\vec a|=\sqrt{a_1^2+a_2^2+a_3^2}$. A unit vector in the direction of $\vec a$ is $\hat a=\dfrac{\vec a}{|\vec a|}$.
Types of vectors
- Zero vector $\vec 0$: magnitude $0$, no definite direction.
- Unit vector: magnitude $1$.
- Equal vectors: same magnitude and direction.
- Collinear/parallel: $\vec a=\lambda\vec b$ for some scalar $\lambda$.
Addition and scalar multiplication
Add componentwise; scalar $\lambda$ multiplies each component (triangle/parallelogram law geometrically). The position vector of a point $P(x,y,z)$ is $x\hat i+y\hat j+z\hat k$, and the vector from $A$ to $B$ is $\overrightarrow{AB}=\vec b-\vec a$ (position vector of $B$ minus that of $A$). The section formula gives the point dividing $AB$ in ratio $m:n$ as $\dfrac{m\vec b+n\vec a}{m+n}$.