Scalars and vectors • Scalars are observables which can be expressed with a simple number (and appropriate units). Examples: Time, mass, temperature,... • Vectors are observables which have both a magnitude (a number with units) and a direction. Examples: Displacement, velocity, force, acceleration… • It is very important to distinguish these two entities: use different notation (A for scalar, A or for vector), and clearly indicate size and direction for a vector result. a ! A
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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.