Scalar Quantities are quantities which represent only magnitude, such as kilograms and metres. This way of presenting quantities only tells you how much of something has happened, it does not tell you in which direction it happened. To solve problems which involve a direction, we use:
Vector Quantities. These are simply distance, speed and acceleration quantities which have a specific direction. Here are some typical vector quantities: 7m [east], 56km/h [north], 4m/s [E56°N].
Lets learn about some useful vector concepts:
Notation:
A vector variable will always have a line/little arrow on top of its variable:
I use the above variable to represent velocity. The vector quantity itself will have its
direction written in square brackets after the magnitude: 75m [east], 33km/h [west].
When the direction does not directly correspond to one of the four directions (NSEW) we can
use angles to describe the direction. The notation that we use is:
Lets analyze a vector quantity, 23m [S55°E].
23m is the displacement, this is the scalar (directionless) portion of the vector quantity.
[S55°E] is the direction portion, which turns this into a vector quantity.
The "S" stands for south, 55 is the angle which the direction takes from south to "E", east.
The actual direction is 55 degrees east of the south direction. The above displacement
can also be written as 23m [E35°S] ( we go from east 35 degrees to the south ).
You already know the most important difference between vectors and scalars - vectors have a direction and scalars don't. However there are other important distinctions.
Displacement is very different than distance; distance is the actual amount of travel that has happened during motion, displacement is how far you are from your reference point. So that if you start off in your home, go around and return you have passed a certain distance, but your displacement is zero because you have returned to your starting point. No matter how you got to a certain point, displacement will always be the most direct distance to that point ( from where you started ). Keep in mind that displacement will also have an angle associated with it.
Velocity is not speed either. Velocity is the change in displacement, so that if you went in zigzags and spent an hour getting to a location 1km away from your starting point then your velocity will be 1km/1hour [direction]. The velocity takes into account only the final position and the total time it got to get there.
You should use vectors any time you use angles and directions of motion, whenever your motion is not forward/backward in the same direction. Vectors are highly useful when they are drawn out, then you can use trigonometric ratios to find out resultant vectors.
When you draw vectors ( which are just a way to describe motion in a visual way ) they look like a line with a "head" and a "tail". The line begins in a tail, which shows where you started from and ends in a head ( which is an arrow ) that shows where the motion ended.
The little d with an arrow on top shows that this vector is a displacement vector, it looks like its in the direction [Right 20° Up] ( we are going 20 degrees Up from the Right direction ). The length of the line is ideally the magnitude of the vector ( the scalar part of it ), so when drawing vectors you should make them reasonably proportional ( e.g. a 3m vector should be smaller than a 7m one ).
An important mathematical tool when using vectors is the absolute value notation. An absolute value of a vector is written like so: || and it is a way to represent just the magnitude of the vector, without direction. We will be taking the absolute value in an example question.
This is the actual application of vectors, adding and subtracting vector quantities. When you have two vectors to add you arrange them head to tail and you draw a line from the first tail to the last head:
The vectors, again, represent displacement. The vectors represent the motion of a person: at first the person moved roughly to the east ( diplacement 1, d1 ), then the person moved northeast (d2) and stopped. Now, to figure out where the person is postioned relative to his/her starting point we need to add the two displacements. We draw a line ( the blue line ) from the starting point to the ending point of the journey, this is the resultant displacement and is labeled .
Addition Example
Q: Billy Bob Joe starts going to school from his house. First he walks north 300m to the bus stop, and then the bus takes him 1km west to school.
a) How far is Mr. Joe from his house when he is at school?
b) When Mr. Joe returns home what will his displacement be?
c) How will Mr. Joe's displacement be affected if he took a different route to school?
A: First lets draw our situation with vectors. Assuming north is up ( when in doubt draw the north arrow ) we draw Mr. Joe's northernly walk and then his westerly ride - a vector going up and a vector going left.
The blue line is the resultant displacement - the final position which Billy Bob Joe has, the tail of the vector is at the house, and its head is at the school.
a) Our vector addition turned out to be a right angle triangle, and
is the hypotenuse. Lets use the pythagorean theorem to find out just how far
Mr. Joe is.
300² x 1000² = ||²
|| = 1044m
Mr. Joe is 1044 metres from his home.
Notice that we have used the absolute value notation || so that we can use the magnitude of without its direction.
b) When Billy Bob Joe returns home his displacement will be zero, he is zero metres away from his home.
c) Taking an alternate route would not change Mr. Joe's displacement, he would still end up in his school. Unlike distance, the displacement is not affected by how you got to a place, its only concerned with the location of that place.
You will very rarely use vector subtraction, although it is very simple: when
subtracting a vector from another vector you are just adding an opposite vector.
1 - 2 = 1 + ( - 2 )
A vector multiplied by a negative number will have its direction reversed ( above
we are multiplying by -1 so its magnitude stays the same ). Visually the vector
which is being subtracted will flip its head and tail, then you rearrange the vectors
head-to-tail again and perform simple addition.
While this section has been an introduction to vectors and slightly theoretical primer you should move on to the next chapter Vectors and Angles to see how useful vectors are when they are combined with angles.