Chapter 1: Vectors

Concept & Definition

Tell someone a car is moving at $60\ \text{km/h}$ and you have only told them how fast. Tell them it is moving at $60\ \text{km/h}$ due north and you have told them what actually matters. Many physical quantities — displacement, velocity, acceleration, force — need this second piece of information, and quantities that carry it are called vectors.

Scalars and Vectors

Physical quantities fall into two families, depending on what it takes to describe them completely:

• A scalar has magnitude only. Mass, time, temperature, distance, and energy are scalars.
• A vector has both magnitude and direction. Displacement, velocity, acceleration, and force are vectors.

We write a vector with an arrow, such as $\vec{v}$, and draw it as an arrow whose length represents the magnitude and whose arrowhead points in the direction. The magnitude alone is written $|\vec{v}|$ and is always a non-negative scalar.

Scalar vs. Vector

A scalar is fully described by a number and a unit. A vector also needs a direction.

Components

A vector drawn on a plane can be split into two perpendicular parts: one along the $x$-axis and one along the $y$-axis. These are its components, $A_x$ and $A_y$. Each component is a signed number — positive when it points along the positive axis, negative when it points against it.

To keep track of direction, we attach a unit vector to each axis: $\hat{i}$ points along $+x$ and $\hat{j}$ along $+y$, each with magnitude one. Multiplying a component by its unit vector turns a plain number back into a vector along that axis.

Key Formula

Vector Components in 2D

$$\vec{A} = A_x\,\hat{i} + A_y\,\hat{j}$$

where $A_x$ and $A_y$ are the components of $\vec{A}$ and $\hat{i}$ and $\hat{j}$ are unit vectors along the $x$- and $y$-axes.

The two components and the vector itself form a right triangle, so the magnitude follows directly from the Pythagorean theorem:

Key Formula

Magnitude of a Vector

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

where $A_x$ and $A_y$ are the components of $\vec{A}$ along the $x$- and $y$-axes.

Direction and Resolution

The same right triangle gives the direction. If $\theta$ is the angle the vector makes with the positive $x$-axis, the components can be recovered from the magnitude and the angle:

Key Formula

Resolving a Vector

$$A_x = |\vec{A}|\cos\theta, \qquad A_y = |\vec{A}|\sin\theta$$

where $|\vec{A}|$ is the magnitude of $\vec{A}$ and $\theta$ is the angle measured counterclockwise from the positive $x$-axis.

Going the other way, the angle is found from the ratio of the components:

Key Formula

Direction of a Vector

$$\theta = \tan^{-1}\!\left(\frac{A_y}{A_x}\right)$$

where $A_x$ and $A_y$ are the components of $\vec{A}$ and $\theta$ is its angle from the positive $x$-axis.

Common Pitfall

A calculator returns $\tan^{-1}$ only between $-90^\circ$ and $90^\circ$, so it cannot tell $(3, 4)$ from $(-3, -4)$. Always check the signs of $A_x$ and $A_y$ to place the vector in the correct quadrant before trusting the angle.

Components let us analyze each direction independently — the idea that makes kinematics and Newton’s laws tractable. A problem in a plane becomes two simpler problems along straight lines.

Quick Check

A vector has components A_x = 3 and A_y = 4. What is its magnitude?

Reveal Explanation

Apply the magnitude formula: $|\vec{A}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.