In this video, we’re going to learn how to represents vectors in the Rectangular and
Polar form.
So, let’s get straight to it!
Alright, to start off, I want to remind you of what vectors are: mathematical quantities
that involve both magnitude and direction.
A good example of a vector quantity is something like “the grocery store is one kilometer
east of here”.
As we can see, this involves the distance, which is the magnitude, as well as the direction.
If this isn’t familiar to you, we encourage you to check out our “Introduction to Vectors”
lesson first before moving on with this one.
Alright, so let’s get a little bit more abstract.
Say we draw some arrows like this.
Now couldn’t we consider these to be vectors too?
They have magnitudes - represented by their lengths - and they point in certain directions.
So yes, they are in fact vectors!
Actually, arrows are a really good way to represent vectors geometrically.
If we wanted to represent wind speed and direction, for example, we could draw an arrow pointing
in the direction of the wind.
We could draw it longer if the wind is stronger and shorter if the wind is lighter.
But we can get even more abstract than that.
Let’s set up some coordinate axes, and draw a random arrow somewhere.
Now let’s put the tail of the arrow where the axes intersect, also known as the origin.
See how the tip of the arrow ends at a certain point?
That point can be described by coordinates, right?
So couldn’t we represent the arrow itself by the coordinates of that point?
Yes, we can!
In fact, we can represent any vector by listing its coordinates!
It works the other way, too.
For any set of coordinates, say (4,2), we can draw an arrow that ends at that point.
So vectors and coordinates on a plot actually correspond to each other.
When a vector is represented by its coordinates, we call that its rectangular representation
or rectangular form.
By the way, this works in 3 dimensions, too!
On a side note, what about the coordinates (0,0)?
We can’t really draw an arrow from a point to itself.
Nonetheless, (0,0) is still considered a vector as well.
It has zero length and an undefined direction.
In mathematics, we call this the zero vector.
Now, in two dimensions, we can represent vectors in another way.
Instead of giving its coordinates, we could specify its magnitude and its angle from a
given axis.
We usually choose the positive ‘x’ axis and measure angles counterclockwise from it,
like this.
So an arrow with length 3 and angle 30 degrees would look like this.
When we represent a vector with a magnitude and an angle, we call that its polar representation
or Polar Form.
Some people write a vector in polar notation like this, with an angle sign preceding the
angle.
But others often write it in a list, which looks very similar to the rectangular representation,
so be careful!
Alright, before we move on, let’s test ourselves with an example.
Which one of these arrows corresponds to a vector with a magnitude of 5 and an angle
of 120 degrees?
Well, the answer is ‘b’.
And how do we know this?
First of all, 120 degrees puts the vector in the second quadrant, which immediately
eliminates the options ‘a’ and ‘d’.
And secondly, between these two, does ‘c’ look like it has a magnitude of 5?
Of course not, it looks more like its magnitude is 1.
Therefore, that leaves us ‘b’ as our final answer!
Awesome!
So let’s move right along.
So, you might be wondering, is it possible to convert between a vector’s rectangular
and polar representations?
Well certainly!
Let’s first look at an example of how to convert a vector from the rectangular form
to the polar form.
Consider the vector (2,2).
We can actually think of this diagram as a right-angled triangle, like so.
In doing so, the magnitude of the vector becomes the length of the hypotenuse of this triangle.
So, we can actually use the Pythagorean theorem to calculate it.
Let’s bring up the formula for it, just to refresh our memory.
Plugging in the respectives values gives us the following.
2 squared plus 2 squared equals 8.
And square rooting both sides gives us a final ‘c’ value or magnitude of square root
of 8 or about 2.83.
As for the angle, we can find that using trigonometry.
Since we know all three sides of this triangle, we can choose any of the trigonometric functions
to solve for the angle.
For instance, the sides opposite to and adjacent to the angle are both 2.
Now, looking at the acronym SOHCAHTOA, we see that tangent is the function that involves
opposite over adjacent!
So the tangent of this angle, which we can just represent as ‘theta’ for now, is
2 divided by 2, which equals 1.
So, all we need to do is take the inverse of that to give us a final answer of 45 degrees
for this angle.
So now we’ve found the equivalent of (2,2) in polar form.
The magnitude is root 8, and the angle is 45 degrees.
Awesome!
But what if we want to go the other way around, from the polar representation to the coordinate
representation?
Well, that’s a matter of simple trigonometry as well.
In fact, for a vector with magnitude ‘r’ and angle theta, the ‘x’ coordinate is
‘r’ times cosine of theta and the ‘y’ coordinate is ‘r’ times sin of theta.
So these are the formulas that give us the conversion!
All we would need to do to solve for them is just plug them into our calculators to
give us the coordinate values.
Pretty simple, right?
Alright, so we’ve reached the end of this lesson!
We looked at how to represent vectors mathematically, in both rectangular form and in polar form,
and learned how to convert between the two.
So make sure to keep practicing and we will see you in the next lesson!