In Class 9 you described motion along a straight line, where direction could be handled with a simple plus or minus sign. Motion in a plane is richer: a cricket ball arcs across the field, a swimmer drifts across a river. To handle two dimensions we need a tool that carries both magnitude and direction together — and that tool is the vector.
Physical quantities fall into two families. A scalar has only magnitude: mass, time, temperature, speed, distance and energy are all scalars and obey ordinary arithmetic. A vector has both magnitude and direction and follows special rules of addition: displacement, velocity, acceleration, force and momentum are vectors. A vector is drawn as an arrow whose length is its magnitude and whose head points in its direction. We write it as $\vec{A}$ and its magnitude as $|\vec{A}|$ or simply $A$.
Two vectors are added head to tail. In the triangle law, the tail of $\vec{B}$ is placed at the head of $\vec{A}$, and the arrow drawn from the tail of $\vec{A}$ to the head of $\vec{B}$ is the resultant $\vec{R}=\vec{A}+\vec{B}$. The parallelogram law states that if two vectors acting at a point are the adjacent sides of a parallelogram, the diagonal through that point is their resultant. When $\vec{A}$ and $\vec{B}$ have an angle $\theta$ between them, the magnitude of the resultant is $R=\sqrt{A^2+B^2+2AB\cos\theta}$, and it makes an angle $\alpha$ with $\vec{A}$ where $\tan\alpha=\dfrac{B\sin\theta}{A+B\cos\theta}$.
Working with arrows geometrically is clumsy, so we resolve each vector into perpendicular components along the x and y axes. Using the unit vectors $\hat{i}$ (along x) and $\hat{j}$ (along y), a vector making angle $\theta$ with the x-axis becomes $\vec{A}=A\cos\theta\,\hat{i}+A\sin\theta\,\hat{j}$. A unit vector has magnitude exactly 1 and only fixes direction; $\hat{A}=\dfrac{\vec{A}}{|\vec{A}|}$. To add vectors by components, simply add the x-parts and the y-parts separately. Two ways of multiplying vectors also appear: the dot (scalar) product $\vec{A}\cdot\vec{B}=AB\cos\theta$ gives a scalar (used for work), while the cross (vector) product $\vec{A}\times\vec{B}=AB\sin\theta\,\hat{n}$ gives a vector perpendicular to both (used for torque).
- Scalar: magnitude only (mass, speed, time, distance).
- Vector: magnitude and direction (displacement, velocity, force).
- Resultant: $R=\sqrt{A^2+B^2+2AB\cos\theta}$ with $\tan\alpha=\dfrac{B\sin\theta}{A+B\cos\theta}$.
- Components: $A_x=A\cos\theta$, $A_y=A\sin\theta$; magnitude $A=\sqrt{A_x^2+A_y^2}$.
- Products: dot $\vec{A}\cdot\vec{B}=AB\cos\theta$ (scalar); cross $|\vec{A}\times\vec{B}|=AB\sin\theta$ (vector).