Scalors
A scalor is a variable that only has a magnitude.
Examples:
| Variable | Magnitude |
|---|---|
| air pressure | 101.3kPa |
| temperature | 21°C |
| price | $50 |
| speed | 10m/s (speed & direction is velocity) |
| distance | 3,000m (distance & direction is displacement) |
Vectors
A vector is a variable that has a magnitude and direction. Temperature is just a scalar. You can't say it is 50° north. Velocity is a vector. You could say the velocity is 50m/s north.
Examples:
| Variable | Magnitude | Direction |
|---|---|---|
| displacement | 10m | West |
| velocity | 20m/s | 20 degrees above the x-axis |
| acceleration | 9.8 m/s² | down |
| displacement | 10m | up |
| velocity | 3m/s | 10 degrees above the horizon |
Vector Components
If we think of a vector like the hypotenuse of a right triangle, then we can use SOH-CAH-TOA to find the lengths of the x and y components of the triangle.
$$ cos \theta = \frac{adjacent}{hypotenuse} $$ $$ (hypotenuse)cos \theta = adjacent$$ $$ (hypotenuse)sin \theta = opposite$$For a velocity vector the hypotenuse is v. The adjacent and opposite sides of the triangle are the x and y part of v.
$$ (v)cos \theta = v_{x}$$ $$ (v)sin \theta = v_{y}$$For displacement the equations are the same
$$ (dist)cos \theta = x$$ $$ (dist)sin \theta = y$$In the simulation below position the mouse to around magnitude 250 and angle 14°. What are the x and y components of the vector?
At what angles are the magnitude and the x-component equal?
What does a negative sign mean for a vector?
Example: A plane is taking off at an angle of 14° above the horizon. If the plane is moving at 250m/s how fast is it moving in only the vertical direction?
solution
$$ (v)sin \theta = v_{y}$$ $$ (250)sin(14) = v_{y}$$ $$ 61 \small\frac{m}{s} = \normalsize v_{y}$$Example:You want to walk to the closest pokemon gym. The compass on your phone says you have to walk north/west. You arrive at the gym after walking 75 meters. Sadly you find that you need to be level 5, but you are level 3. How far west did you walk?
solution
$$ (v)cos \theta = x$$ $$ (75)cos(45) = x$$ $$ 53m = x$$Example:You walk 3 miles north and then 4 miles west. How far away are you from your starting location?