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Vectors

In the past few weeks I have talked about different quantities in physics. We have tried to categorize these quantities according to different criteria. In the study of motion it makes sense to introduce another way to differentiate between quantities. In this chapter I will talk about how physical quantities can be divided up into tow categories: vectors and scalars.

Thus far we have only talked about scalars, although at times you have asked me whether speed for instance is the same as velocity. Now we can distinguish the two - one is a vector the other a scalar. Scalars are quantities that can be described just with numbers (and a unit). We call this number the magnitude (or size). Vectors on the other hand have in addition to the magnitude, the umber, also a direction. Mass is a scalar. Five kilograms are just that, 5 kg. However, if you are going 50 mph it matter in which direction you are going. Depending on the direction you could either end up in New York or n Boston - 50 mph East is a vector. It is a quantity that has both a magnitude and a direction. We call this quantity the velocity, as opposed to speed, which is a scalar. Other vectors are force or displacement. If you are asking for direction you are asking to be given a vector, go this far in this direction, then turn left and go this far n the new direction and so on.

 Vectors are represented simply as arrows. This makes sense since arrows have both a magnitude (size) and a direction. All arrows pointing in the same direction and with the same length are identical. Just like all numbers 3 are identical, regardless as to where you write them down. When you ask for directions, in your mind you have to put one of these vectors behind the other to make sense of the instruction. This is called adding the vectors together. The easiest adding of vectors would be: first you go 5 miles West and then (after picking something up for instance) you go another 3 miles in the same direction. You will have covered 8 miles overall. Since the two vectors are in the same direction (called co-linear) they add just like numbers.

When this is graphically displayed the vectors are put head-to tail. Imagine you are walking on a train. The train is going 20 m/s to the North and you walk to front of the train at a speed of 3 m/s. Your total speed with respect to the outside is now 23 m/s. The vectors add simply like numbers.

 If the vectors are not co-linear things are a little more complicated. Imagine you are trying to cross a river. The river flows with 3 m/s to the left and you are trying to cross by swimming at a steady pace of 4 m/s. You know from experience that you are going to end up somewhere downstream. How far depends on the width of the river. How much will your total speed be? It would be 7 m/s if you are going downstream, but has to be less since you are swimming across. The answer the question can be found in the Pythagorean Theorem. In order to find your total velocity vector, the speed and the new direction you have to add the vectors like vectors not like numbers.
 You have to take one of the vectors and put it's tail of the head of the other. The vector sum then goes from the tail of the first to the head of the second vector. The direction is therefore half left, but what about the magnitude? That can be found using Pythagorean Theorem, that says that the length of the hypotenuse of a right triangle can be found by taking the square root of the sum of the squares of the other side - in short Click here for more excercises and an applet for adding vectors.

Vector subtraction can be explained in terms of vector addition, much like when you were first taught how to subtract. Your teacher did not ask you: "How much is 7 minus 4?" Instead she explained to you that you need to find that number that has to be added to 4 to get 7.

Vector A is added to vector B to get vector C (in red). A + B = C (Vectors are written in bold).

This can be changed to: A - C = B. B is therefore the result of subtracting A and C. Which gives us the rue that vectors are subtracted by putting them tail to tail and connecting the heads. The difference points to the second vector.

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