Vectors

We draw a vector $\vec A$ as an arrow. The arrow points in the vector's direction, and its length shows the vector's magnitude, written $|\vec A|$ or just $A$. Where the arrow is drawn does not matter. Two arrows with the same length and direction are the same vector, wherever they sit.

Vectors come with two operations.

Addition. To add $\vec B$ to $\vec A$, place the tail of $\vec B$ at the tip of $\vec A$. The sum $\vec C = \vec A + \vec B$ is the arrow from the tail of $\vec A$ to the tip of $\vec B$. You can also draw both arrows from the same point and complete the parallelogram; the diagonal is the same $\vec C$.

The order does not matter (adding is commutative), and when you add three vectors it does not matter which pair you add first (it is associative):

There is a zero vector $\vec 0$ that changes nothing when added, and every vector has an opposite, $-\vec A$: same length, reversed direction, so that $\vec A + (-\vec A) = \vec 0$. Subtracting a vector means adding its opposite.

Multiplication by a number. Multiplying $\vec A$ by a number $c$ stretches its length by $|c|$. If $c$ is positive the direction stays the same; if $c$ is negative the direction flips. Dividing a vector by its own magnitude leaves an arrow of length exactly 1 that points the same way. This is a unit vector, marked with a hat:

Drag the arrows in the simulation, add them, and watch the components and the sum change. Compare what you see with the rules above.

Arrows are good for intuition, but to calculate we need numbers. A coordinate system turns vectors into numbers. Setting one up takes four choices:

1. an origin, the point every position is measured from;
2. a set of axes;
3. a positive direction on each axis;
4. a set of unit vectors at every point.

These choices are ours to make. A good choice can turn a messy problem into an easy one, which is why it pays to draw the coordinate system before calculating anything.

Cartesian coordinates use three perpendicular axes $x$, $y$, $z$. At every point the unit vectors $\hat i$, $\hat j$, $\hat k$ point toward increasing $x$, $y$ and $z$. They point the same way everywhere, which makes Cartesian coordinates the simplest system to calculate in.

Polar coordinates describe a point in a plane by its distance $r$ from the origin and the angle $\theta$ from the $+x$ axis, measured counterclockwise. Adding a height $z$ gives cylindrical coordinates. The two descriptions are linked by

(The inverse tangent needs care about which quadrant the point is in; more on that below.) Polar coordinates also have their own unit vectors: $\hat r$ points straight away from the origin and $\hat\theta$ points around the circle, counterclockwise. Unlike $\hat i$ and $\hat j$, these change direction from point to point. That will matter when we study circular motion.

Once we have unit vectors, any vector can be split into pieces along the axes. The numbers in front of the unit vectors are the components:

A component is a signed number: $A_x$ is negative if $\vec A$ points partly toward $-x$. So one vector in three dimensions is described by three numbers $(A_x, A_y, A_z)$.

Magnitude. By Pythagoras, the length of the arrow is

Direction. For a vector in the $xy$-plane that makes an angle $\theta$ with the $+x$ axis,

Adding arrows head to tail is exactly the same as adding their components separately:

That is the real reason components are useful. Adding arrows takes geometry; adding numbers takes arithmetic.

• A vector has a magnitude and a direction; it adds head to tail, and multiplying by a number stretches or flips it.
• A coordinate system needs an origin, axes, positive directions and unit vectors.
• In components, $\vec A = A_x\hat i + A_y\hat j + A_z\hat k$, with $A = \sqrt{A_x^2 + A_y^2 + A_z^2}$.
• Vectors add component by component. Watch the quadrant when you convert components back into an angle.