Vectors

Written by tutor Megan C.

When working with equations and mathematical operations with scalar quantities,
you are looking at only the magnitude of the numbers. Therefore, when you solve
a problem and find a numerical answer, you have the size of the answer but not its
position. Using vectors instead of simple numbers can help you solve problems where
you need more information. A vector is a number with a direction.

For example, say you are given the speed of an object (2 m/s) to use in a problem.
Speed is a scalar quantity, so you only have the magnitude of the speed. If a problem
gives you a speed and its direction (2 m/s west), you have velocity, which is a vector
quantity and you can use this to find significantly more information.

There are a lot of different ways to express vector quantities. Using the cartesian
coordinate system, you can express vectors in terms of x, y, and z. That would mean
a velocity could be 2m/s in the x-direction, 1m/s in the y-direction, and 3m/s in the z-
direction. Sometimes you may see this expressed using a unit vector. A unit vector has
a magnitude of 1 in either the x, y, or z direction. The x unit vector is i, the y unit vector
is j, and the z unit vector is k. So, the velocity we just described would be 2i+1j+3k m/
s. Vectors may also be expressed using polar coordinates. This would mean that your
quantity would be expressed using a radius and angle. If our velocity was given as 2 m/
s at 30°, it would be expressed as 2r m/s. In this case, r changes when the polar angle
changes.

It is usually easier to work with vectors in cartesian coordinates, because once we
establish our coordinate system, we have a set i, j, and k that will be constant for
any vectors we are working with. That means that any vector in the system will have
an x,y, and z component (even if they are 0). To add vectors, you just add the x-
components, y-components, and z-components. So to add our cartesian velocity from
the last paragraph (2i+1j+3k m/s) to another velocity (3i+2j+1k m/s), we just add the
components to get 5i+3j+4k m/s. The same process can be used to subtract one vector
from another.

Vector equilibrium occurs when the sum of all vectors in a system is zero. When you
have a system of two vectors, this is really easy. All you need are two vectors with the
same magnitude and opposite directions. For example, the vectors 5i and -5i would be
in equilibrium. It gets more complicated when you have more than two vectors.
Let’s say you are given two vectors and you need to find the third vector that will put the
system in equilibrium. The third vector is called the equilibrant. A good way to find this
third vector is to figure out the sum of the first two vectors. This will give you essentially
one vector, turning this problem into an easy two-vector problem. The equilibrant vector
would be the same magnitude of the resultant vector in the opposite direction. The
following is what this would look like if you drew it:

The same process can be used for systems with many vectors. Just add all of the
vectors together, and the equilibrant will be the same magnitude in the opposite
direction of the sum of the vectors.

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