Vectors

Physical quantities come in two flavors. A scalar is described completely by a single number with a unit — mass (2.5 kg), temperature (300 K), speed (12 m/s). A vector requires both a magnitude and a direction — displacement, velocity, force, and acceleration are all vectors.

On paper we draw vectors as arrows. The length of the arrow represents the magnitude; the arrowhead shows the direction. We write vectors in boldface (A) or with an overhead arrow ($\vec{A}$). The magnitude is written $|\vec{A}|$ or simply $A$ (italic, no arrow).

Two vectors are equal if and only if they have the same magnitude and the same direction — regardless of where their tails are placed. This means you can slide a vector arrow anywhere in space without changing it, as long as you keep its length and direction the same.

The geometric rule for adding two vectors is the head-to-tail method: place the tail of $\vec{B}$ at the head (tip) of $\vec{A}$. The resultant $\vec{R} = \vec{A} + \vec{B}$ is the arrow drawn from the tail of $\vec{A}$ to the head of $\vec{B}$.

To subtract a vector, reverse its direction: $\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$. The vector $-\vec{B}$ has the same magnitude as $\vec{B}$ but points the opposite way.

For more than two vectors, simply continue the head-to-tail chain. The resultant always runs from the very first tail to the very last head.

Any 2-D vector $\vec{A}$ can be "projected" onto perpendicular axes. The x-component is the shadow it casts on the horizontal axis; the y-component is its shadow on the vertical axis. Together they completely specify the vector.

If $\vec{A}$ makes an angle $\theta$ with the positive x-axis, then:

And to go the other way — from components back to magnitude and direction:

The component method turns every vector-addition problem into simple arithmetic. The rule: add x-components together, add y-components together.

For $\vec{R} = \vec{A} + \vec{B}$:

Then recover the resultant's magnitude and direction:

Worked Example

Let $\vec{A} = (3,\,4)$ and $\vec{B} = (5,\,-2)$ (components in meters). Find $\vec{R} = \vec{A} + \vec{B}$.

A unit vector has magnitude exactly 1. Its only job is to specify a direction. The standard Cartesian unit vectors are:

Any 3-D vector can be written as a sum of scaled unit vectors:

This notation is powerful because it makes arithmetic obvious: add the $\hat{i}$ terms together, the $\hat{j}$ terms together, and the $\hat{k}$ terms together.

Multiplying two vectors together can produce either a scalar or a vector, depending on which multiplication rule you use. The dot product (or scalar product) gives a scalar:

where $\theta$ is the angle between the vectors and $A$, $B$ are their magnitudes. Equivalently, in component form:

When $\theta = 0°$ (parallel) the dot product is maximum $= AB$. When $\theta = 90°$ (perpendicular) it is zero. When $\theta = 180°$ (antiparallel) it is $-AB$.

The cross product $\vec{A}\times\vec{B}$ produces a vector that is perpendicular to both $\vec{A}$ and $\vec{B}$. Its magnitude is:

The direction is given by the right-hand rule: point your fingers along $\vec{A}$, curl them toward $\vec{B}$, and your thumb points in the direction of $\vec{A}\times\vec{B}$.

In component form the cross product is computed from a determinant-like expansion:

The cross product appears throughout physics: torque is $\vec{\tau} = \vec{r}\times\vec{F}$, angular momentum is $\vec{L} = \vec{r}\times\vec{p}$, and the magnetic force on a charge is $\vec{F} = q\vec{v}\times\vec{B}$.

Most physics problems involving multiple forces, velocities, or displacements reduce to a sequence of vector operations. Here is the strategy that works every time:

Mastering vectors now pays dividends in every topic that follows — 2D kinematics, Newton's second law in two dimensions, work and energy, torque, and fields all depend on fluent vector algebra.