Lesson - Vectors

When we apply mathematics to the real world, sometimes it is not enough just to use a number. If you want to describe where America is in relation to Chile, Japan, or Israel, you need to know not just how far away the other countries are, but in what direction they are.

Or, for that matter, driving your car faster won't get you to your destination any sooner if you are driving the wrong way.

The world we exist in is three dimensional (four if you count time), and thus the quantities that express the real world need some ability to deal with this. We need to be able to express not just magnitude (how fast, how far, how strong of a push) but also direction.

Often this is done by expressing the direction as an angle measured from some fixed standard. A plane might be heading 30 degrees west of north, for example. Since in mathematics we often like to express things with respect to the x-y axes, often direction is expressed as an angle counterclockwise from the positive x axis.

Not all quantities have to have a direction, however. In what direction is temperature, for example. Or sometimes, if we just have the magnitude, we treat it as a scalar. A car's speed is just that, a scalar measure of how fast the car is moving with respect to its surroundings. A car's velocity, on the other hand, is a vector measure of the cars speed in a given direction.

1. For the following quantities, determine which are vectors and which are scalars.
   • position
   • length
   • force
   • temperature
   • torque (the twisting force applied when you tighten a bolt or loosen a twist-off bottle cap.)
   • price

Sometimes, however, discussing a vector quantity in terms of magnitude and direction isn't very useful. Suppose you are giving directions in a city block. Saying a building is 2 times the square root of 2 blocks in a direction 45 degrees west of north may not be as useful as saying a building is 2 blocks north and 2 blocks west.

In mathematics, what we usually like to use is x and y coordinates (x, y, and z if we have three dimensions). A given vector can be converted from magnitude and direction to coordinates by considering what would happen if we drew an arrow starting from the origin (0,0) in the vector's direction with a length equal to the vector's magnitude.

2. For the following vectors, the variable r indicates the magnitude, and the variable q indicates direction as angles measured in degrees counterclockwise from the positive x axis. Tell whether the x coordinate is positive, zero, or negative. Do the same for the y coordinate.
   • r=3, q=45
   • r=2, q=120
   • r=5, q=180
   • r=1, q=200
   • r=1, q=300
3. For the following vectors, convert from magnitude and direction to x and y coordinates.
   • r=3, q=180
   • r=2, q=195
   • r=5, q=45
   • r=1, q=120
   • r=1, q=323
4. For the following vectors, convert from x and y coordinates to magnitude and direction.
   • x=2, y=2
   • x=1, y=-1
   • x=3, y=-2
   • x=0, y=-3
   • x=-2, y=1