What are vectors?

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 #-pi/2# and #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 = mcostheta#
#y = msintheta#

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.