How to Add Different Vectors

Back to Maths Grade 9/10


In the last tutorial we covered how to add simple vectors. 'Simple' vectors are either pointing in the exact same direction or in the exact opposite direction; in these cases you can rely on numbers and signs to work out the answer. But what happens when you have vectors pointing in different directions altogether - say, one pointed to the North, and another pointed to the South East or even just easterly?

Let's work through an example.
An Example
Imagine a boat cruising through the seas: the boat leaves your home city (A) and travels 20 km north to island B. On the journey home, the captain decides to take its occupants home via another island (C), which is 20 km east of island B. What vector describes the position (or displacement) of the boat at island C, in relation to your home city (A)?

The Solution
Step 1: Draw the problem using Arrows.
Being able to depict vectors using arrows is really useful for learning how to add or subtract different vectors. Before we draw the vectors for our example problem, let's cover a bit of basic vector naming convention. An arrow vector has a starting point and an end tip.

When you use an arrow to depict a journey, the end tip always points in the direction of the vector - you probably already know this intuitively. The starting point is always positioned at the beginning of the journey, wherever that may be in your picture / on your page. If the problem you're working on isn't about a journey, but just about adding vectors in general, you must make sure to join the end tip of one vector to the starting point of the next one in the problem - and continue to do that all of the vectors outlined in the problem question.

You'll see there are three vectors in the image below (mouse over to see the islands). The first one illustrates the journey from your home city (A) to island B; we can call this vector AB (it starts at point A, so A comes first in the naming). The second vector depicts the journey from island B to island C; we can call this one vector BC. The last vector depicts the displacement of the boat at island C, in relation to your home city (A); we can call this vector AC - it is the vector of interest. It tells us how far and in what direction a boat would need to travel if it was going to go directly from your home city (A) to island C.

Step 2: Work out the Magnitude of the Vector of Interest
Using trigonometry you can work out the magnitude of the vector of interest - in this instance you'll use pythagorus' theorum.

a2 + b2 = c2
AB2 + BC2 = AC2
202 + 202 = AC2
400 + 400 = AC2
800 = AC2
√800 = √AC2
28.3 km = AC

Step 3: Work out the Direction
To work out what direction a boat would need to head out in from your home city (A) to travel directly to island C, we need to work out what angle CAB is. If you've drawn your problem to scale (as above), you can use a protractor to work out the angle CAB. The alternative is you can use mathematics to work out the angle, which we will do below.

The triangle has two sides that are the same length, as well as a ninety degree angle; this makes it a right-angled isosceles triangle.

There are 180 degrees in a triangle. If one of them is 90 degrees, there is 90 degrees to divvy between the two remaining angles (CAB and BCA). As the angles are equal angles, we can simply divide ninety degrees by two. Ninety degrees divided by two equal angles means that each angle is forty-five degrees. So, our angle of interest is forty-five degrees. We need to give this angle some perspective in relation to a compass; it is forty-five degrees from North, towards the East i.e. 45 degrees North East.

Step 4: Our Answer is...
The vector of interest is 28.3 km, 45 degrees North East.

Some tips

Some tips to summarise: