Scalars and vectors

• Scalars are observables which can be expressed with asimple number (and appropriate units). Examples: Time,mass, temperature,...

• Vectors are observables which have both a magnitude (anumber with units) and a direction. Examples:Displacement, velocity, force, acceleration…

• It is very important to distinguish these two entities: usedifferent notation (A for scalar, A or for vector), andclearly indicate size and direction for a vector result.

a

!

! A

Example: Displacement

• Size: actual distance from Point A to Point B (don’t forgetunits)

• Direction: Describe which way to go.• Note: Vectors which have different sizes, different units, or

different directions are different. BUT: starting point doesnot matter:

Specifying a vector

1.) By giving its length (size, magnitude, absolute value -with units) and its direction.Examples: “1100 m exactly northeast from here” or “0.17m at an angle of 45o above the x-axis, in the x-y plane”.

2.) You can represent vectors by drawing arrows. The lengthof the arrow represents the size of the vector (e.g., 2 cmrepresent 2 N) and the direction of the arrow is in thedirection of the vector.

Sebastian Kuhn:See YF slide

Vectors can be added to (or subtractedfrom) each other.

1.) Geometrically: See examples. Use either “tail to head”method or “parallelogram” method

2.) Mathematically (a bit too advanced for our purpose)

Example: Equilibrium - car at rest• Equilibrium: All forces acting on an object add up to zero

(vectorially).• The object will either be (stay) at rest or will move with

constant velocity.• Example: Car sitting still on an inclined plane (or moving

down with constant velocity)

α

F NormalF Friction

F Weight

xy

Example: Car accelerating down ramp• Net force: All forces acting on an object add up to a net

force along the ramp (vectorially).• The object will accelerate down the ramp.

α

F Normal

F Friction

F Weight

x y

Equilibrium - Sailboat

Normal Forceon KeelDrag

Wind Force