Lecture 3 - Coordinate Systems Flashcards
define: coordinate system
a scheme for representing things as sets of numbers, or coordinates
How many coordinates do you need to represent something?
as many as that thing’s DOF
location is usually represented using ____ coordinates
Cartesian
Cartesian coordinates
vectors i, j, k
Using the Cartesian system, any location realtive to the origin can be expressed uniquely as a…
weighted sum of the vectors
v = v1i + v2j + v3k
What are the coordinates of vector v?
v = (v1, v2, v3)
Right-handed Cartesian coordinates
index finger = i
middle finger = j
thumb = k
How to calculate the sum of 2 vectors, w = v + u?
w1 = u1 + v1
w2 = u2 + v2
w3 = u3 + v3
how to calculate vector length?
|v| = (v12 + v22 + v32)1/2
Pythagoras’s formula
What does the dot product measure?
how close to paralle 2 vectors are
Formula for dot product
u • v = u1v1 + u2v2 + u3v3
u • v = |u| |v| cosø (angle between u and v)
Dot product: 2 vectors →
number
Formula for cross product
u x v = (u2v3 - u3v2, u3v1 - u1v3, u1v2 - u2v1)
Cross product: 2 vectors →
vector
the cross product is non- ______
commutative (order matters)
u x v!= v x u
Length of cross product u x v
u x v | = |u| |v| |sinø| (angle between u and v)