If you wish to assimilate the knowledge in this node fully, you will benefit from
graph paper and an understanding of the knowledge ensconced
here.
Draw two lines, one from (1,1) to (1,3) and one from (1,3) to (2,5). The vector which represents the first line is (0,2), and the one for the second is (1,2).
Now draw a line from (1,1) to (2,5). What's the vector representing that line? It's (1,4), which happens to be (1,2) + (0,2). This is very logical when you think about it - the vector from (1,1) to (2,5), which we call the resultant vector of the other two, represents the entire "journey" made by the other two vectors - they are merely components of it.
This relationship can come in very useful (admittedly, mainly in math classes). You can learn a lot about the vector triangles you've been drawing by using geometry and trigonometry. It's possible to calculate the resultant or either of the other two vectors by knowing the other two. If you know the resultant and one of the others, then simply subtracting the other from the resultant will give you the remaining vector.
Vectors are often used in velocity calculations. A system where velocity is at a constant, and then is effected and changed by some other factor is a consistent player in maths questions. The "riverboat question" comes up again and again.
In this scenario, a boat is said to be sailing through the water at a velocity of, say, (0,-3) - directly downwards. The velocity of the water is said to be, say, (-5,0) - directly right. What is the resultant velocity of the boat?
Easy. Resultant velocity = (0,-3) + (-5,0) = (-5,-3). We can use the techniques we learnt earlier to work out the magnitude of this resultant velocity vector, which is the speed -
h2 = a2 + b2
h2 = -52 + -32
h = √34
h = 5.8 km h-1