Vectors & Scalars


A scalar is a variable that only has a magnitude.

Variable Magnitude
air pressure 101.3 kPa
temperature 21° C
price $50
speed 10 m/s
distance 3,000 m


A vector is a variable that has a magnitude and direction. Temperature is just a scalar with no direction. You can't say it is 50°C down. Velocity is a vector. You could say the velocity is 50m/s down.

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

Vector Components

c a b

Think of a vector like the hypotenuse of a right triangle. This makes the other two sides of the triangle the horizontal and vertical components of the vector.

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

$$a^{2} + b^{2} = c^{2}$$

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

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

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

For displacement the equations are the same

d x y θ
$$ x = (d) \cos \theta $$ $$ y = (d) \sin \theta $$

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 $$ v_{y} = (v) \sin \theta $$ $$ v_{y} = (250 \, \mathrm{\tfrac{m}{s}}) \sin(14 \degree) $$ $$ v_{y} = 61 \mathrm{\tfrac{m}{s}} $$
Example: You want to walk to the closest pokémon gym. The compass on your phone says you have to walk northwest. 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?

northwest means 45 degrees

$$ x = (v) \cos \theta $$ $$ x = (75 \, \mathrm{m}) \cos(45 \degree) $$ $$ x = 53 \, \mathrm{m} $$
Example: You walk 3 miles north and then 4 miles west. How far away are you from your starting location?
solution $$a^{2}+b^{2} = c^{2}$$ $$3^{2}+4^{2} = c^{2}$$ $$25 = c^{2}$$ $$5 \, \mathrm{miles} = c$$