What are vectors?

May 15, 2014

A vector is a quantity that has both a magnitude and a direction.

An example of a vector quantity could be an object's velocity. If an object is moving at 10 meters per second East, then the magnitude of its velocity is 10 m/s, and its direction is East. Direction can be indicated however you'd like, but usually it's measured as an angle in degrees or radians.

Two-dimensional vectors are sometimes written in unit vector notation. If we have a vector $\vec{v}$, then it can be expressed in unit vector notation as:

vec v = x hat ı + y hat ȷ

Think of $\vec{v}$ as a point on a graph. $x$ is its position along the x-axis, and $y$ is its position along the y-axis. hat ı simply indicates the component in the horizontal direction, and hat ȷ indicates the component along the vertical.

To illustrate this, let's say we have a vector vec v = 3 hat ı + 2 hat ȷ. The total magnitude, $m$, of this vector is the length of the line you see drawn from the origin to (3, 2). This magnitude is easy to find; just use the Pythagorean theorem:

m = sqrt(x^2 + y^2) = sqrt(3^2 + 2^2) = sqrt(13) ≈ 3.61

If you're looking to find the direction of this vector, solve for the angle between the x-axis and the vector line. Since this vector ends up in the first quadrant, we can find its direction simply with:

theta = arctan(y/x) = arctan(2/3) ≈ 33.69°

However, be careful when finding the angle... arc tangent always gives a measurement between $- \frac{\pi}{2}$ and $\frac{\pi}{2}$. Make sure you use the correct values for $x$ and $y$, and add the resulting angles correctly.

$x$ and $y$ can also be written in terms of $m$ and $\theta$:

$x = m \cos \theta$
$y = m \sin \theta$

This is useful for when you know a vector's magnitude and direction and want to write it in unit vector form, or for when you're solving projectile motion problems.