Spring 2016, CS-116B Lecture: A review of classical mechanics in physics

A review of classical mechanics in physics
CS-116B: Graphics Algorithms
Instructor: Rob Bruce
Spring 2016

SLIDE 1: Newton's three laws of motion

1st law of motion (law of inertia): A body remains at rest, or moves in a straight line (at a constant velocity), unless acted upon by a net outside force.
2nd law of motion: The acceleration of an object is proportional to the force acting upon it. F = m * a
3rd law of motion (law of reciprocal actions): For every action, there is an equal and opposite reaction.

SLIDE 2: Vectors

A vector contains two components:

A scaler value
A direction

Example: vector v = Aî + Bĵ + Ck̂

Thee-tuple unit vector illustration on a three-dimensional Cartesian coordinate system

Image source: https://upload.wikimedia.org/wikibooks/en/9/98/Unit-vectors-in-Cartesian-Coord.png

SLIDE 3: Computing vector magnitude

vector v = Aî + Bĵ + Ck̂

Magnitude of vector v:

||v|| = SQRT(A² + B² + C²)

SLIDE 4: Normalizing a vector (also called unit vector)

vector v = Aî + Bĵ + Ck̂

normalized vector v = v / ||v|| = v / SQRT(A² + B² + C²)

SLIDE 5: Computing the vector dot product (also called scalar product)

vector v₁ = Aî + Bĵ + Ck̂

vector v₂ = Dî + Eĵ + Fk̂

v₁ • v₂ = ||v₁|| * ||v₂|| * cos(θ)

v₁ • v₂ = Aî • Dî + Aî • Eĵ + Aî • Fk̂ + Bĵ • Dî + Bĵ • Eĵ + Bĵ • Fk̂ + Ck̂ • Dî + Ck̂ • Eĵ + Ck̂ • Fk̂

v₁ • v₂ = A*D*cos(0°) + A*E*cos(90°) + A*F*cos(90°) + B*D*cos(90°) + B*E*cos(0°) + B*F*cos(90°) + C*D*cos(90°) + C*E*cos(90°) + C*F*cos(0°)

v₁ • v₂ = A*D + B*E + C*F

SLIDE 6: Computing the vector cross product (also called vector product)

The right hand rule:

î × ĵ = k̂

ĵ × k̂ = î

k̂ × î = ĵ

î × k̂ = -ĵ

ĵ × î = -k̂

k̂ × ĵ = -î

vector v₁ = Aî + Bĵ + Ck̂

vector v₂ = Dî + Eĵ + Fk̂

n̂ represents the unit vector perpendicular to vectors v₁ and v₂ using the right hand rule.

v₁ × v₂ = ||v₁|| * ||v₂|| * sin(θ) n̂

v₁ × v₂ = Aî × Dî + Aî × Eĵ + Aî × Fk̂ + Bĵ × Dî + Bĵ × Eĵ + Bĵ × Fk̂ + Ck̂ × Dî + Ck̂ × Eĵ + Ck̂ × Fk̂

v₁ × v₂ = A*D*sin(0°) + A*E*sin(90°)k̂ - A*F*sin(90°)ĵ - B*D*sin(90°)k̂ + B*E*sin(0°) + B*F*sin(90°)î + C*D*sin(90°)ĵ - C*E*sin(90°)î + C*F*sin(0°)

v₁ × v₂ = A*Ek̂ - A*Fĵ - B*Dk̂ + B*Fî + C*Dĵ - C*Eî

v₁ × v₂ = (B*F - C*E)î + (C*D - A*F)ĵ + (A*E - B*D)k̂