Vectors & Scalars

Our universe has three dimensions of space and one dimension of time. The three different spacial dimensions make communicating information about space ambiguous.

y x z

Information about the three dimensions of space needs a direction in addition to the magnitude. The pairing of direction and magnitude is called a vector.

If I say "bike 10 km to get to the store", you still can't find the store because you don't know the direction. I should say "bike 10 km West".

Scalars

A scalar is a variable that has a magnitude, but not a direction in space. Temperature is a scalar. You couldn't say it is 50°C to the left.

Variable Magnitude
air pressure 101.3 kPa
temperature 21° C
price $50
speed 10 m/s
distance 3000 m

Speed and distance seem like they could have a direction, but they are defined as only the magnitude of velocity and displacement vectors.

Vectors →

A vector is a variable that has a magnitude and direction in space. Velocity is a vector. You could say a velocity is 10 m/s in the west direction.


Variable Magnitude Direction
displacement 10 m West
displacement 5.5 m up
velocity 20 m/s 20° above the x-axis
velocity 3 m/s 10 left
acceleration 9.8 m/s²

Think of a vector like the hypotenuse of a right triangle. This makes the other two sides of the triangle the components of the vector. Often the components are horizontal and vertical, but they don't have to be.

c a b

We can use the pythagorean theorem to solve for the magnitude of the sides.

$$a^{2} + b^{2} = c^{2}$$ Example: You walk 3 miles north and then 4 miles west. How far away are you from your starting location?
solution $$a^{2}+b^{2} = c^{2}$$ $$3^{2}+4^{2} = c^{2}$$ $$25 = c^{2}$$ $$5 \, \mathrm{miles} = c$$

Vectors and Angles

We can use trig functions (SOH-CAH-TOA) to find how the vector components relate to the angle of the vector.

$$ \sin \theta = \frac{\mathrm{opp}}{\mathrm{hyp}} $$ $$ \cos \theta = \frac{\mathrm{adj}}{\mathrm{hyp}} $$ $$ \tan \theta = \frac{\mathrm{opp}}{\mathrm{adj}} $$
hypotenuse adjacent opposite θ
$$ \mathrm{adj} = (\mathrm{hyp}) \cos \theta \quad \quad \mathrm{opp} = (\mathrm{hyp}) \sin \theta $$

For a velocity vector the hypotenuse is v. The adjacent and opposite sides of the triangle are the x and y part of v.

$$ v_{x} = v\,\cos \theta$$ $$ v_{y} = v\,\sin \theta $$
v Vx Vy θ

For displacement the equations are the same

$$ x = d\,\cos \theta $$ $$ y = d\,\sin \theta $$
d x y θ

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 250 m/s how fast is it moving in only the vertical direction?
solution $$ v_{y} = (v) \sin \theta $$ $$ v_{y} = (250 \, \mathrm{\tfrac{m}{s}}) \sin(14 \degree) $$ $$ v_{y} = 61 \mathrm{\tfrac{m}{s}} $$
0 10 20 30 40 50 60 70 80 90 350 340 330 320 310 300 290 280 270 180 170 160 150 140 130 120 110 100 190 200 210 220 230 240 250 260 N W E S NW SW NE SE Example: You want to walk to the closest pokémon gym. The compass on your phone says you have to walk northeast. 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 north did you walk?
solution

Northeast is an angle of 45 degrees to the north.

0 10 20 30 40 50 60 70 80 90 350 340 330 320 310 300 290 280 270 180 170 160 150 140 130 120 110 100 190 200 210 220 230 240 250 260 N W E S NW SW NE SE $$ d_\mathrm{north} = d \cos \theta $$ $$ d_\mathrm{north} = (75 \, \mathrm{m}) \cos(45 \degree) $$ $$ d_\mathrm{north} = 53 \, \mathrm{m} $$