Motion in a Plane - Scalars and Vectors

Scalars are physical quantities that have only magnitude and no direction. Examples include mass ($m$), time ($t$), and temperature ($T$).

Vectors are physical quantities that have both magnitude and direction and obey the laws of vector addition. Examples include displacement ($\vec{s}$), velocity ($\vec{v}$), and force ($\vec{F}$).

A Unit Vector is a vector of unit magnitude in a specific direction, defined as $\hat{A} = \frac{\vec{A}}{|\vec{A}|}$. The unit vectors along the $x, y, z$ axes are $\hat{i}, \hat{j}, \hat{k}$ respectively.

The Triangle Law of Vector Addition states that if two vectors are represented by two sides of a triangle taken in the same order, their resultant is represented by the third side taken in the opposite order: $\vec{R} = \vec{A} + \vec{B}$.

The Parallelogram Law of Vector Addition states that if two vectors acting at a point are represented by the adjacent sides of a parallelogram, the diagonal passing through their point of intersection represents the resultant.

Resolution of a vector involves splitting a vector into its components. For a vector $\vec{A}$ making an angle $\theta$ with the $x$-axis, the components are $A_x = A \cos \theta$ and $A_y = A \sin \theta$.

The Scalar (Dot) Product of two vectors $\vec{A}$ and $\vec{B}$ is a scalar quantity defined as $\vec{A} \cdot \vec{B} = AB \cos \theta$. It is used to find the work done $W = \vec{F} \cdot \vec{d}$.

The Vector (Cross) Product of two vectors $\vec{A}$ and $\vec{B}$ is a vector $\vec{C} = \vec{A} \times \vec{B}$ with magnitude $AB \sin \theta$ and direction perpendicular to the plane containing both vectors, determined by the Right-Hand Thumb Rule.