Chapter 1: Vectors

Many physical quantities need more than a number. To describe a displacement, velocity, force, or acceleration, we must also say which way it points. These quantities are vectors.

Scalars and vectors

A scalar has magnitude only. Mass, time, temperature, and energy are scalars.

A vector has magnitude and direction. We write a vector in bold, such as $\mathbf{v}$, or draw it as an arrow. Its length represents magnitude and its arrowhead represents direction.

Components

On a two-dimensional coordinate system, a vector can be expressed using horizontal and vertical components:

$$\mathbf{A} = A_x\mathbf{i} + A_y\mathbf{j}$$

where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors along the $x$- and $y$-axes. The magnitude of $\mathbf{A}$ is

$$|\mathbf{A}| = \sqrt{A_x^2 + A_y^2}.$$

Components let us analyze each direction independently, which is essential for kinematics and Newton’s laws.

Next, study Chapter 2: Kinematics.