5 - How to Create a Cartesian System
The diagrams shows two sliders in the top right corner to represent the values a and b, such that a, b increase by increments of 0.1.
a, b are rational numbers
5.1 - Adding scaled vectors
Figure 5.1
Figure shows slider a and b
Vector u and v are given
the construction shows, vectors a*u and b*v head to tail
vector w = a*u + b*v
this shows that in the 2d plane, we can create any vector w, being compose of two scale non parallel vector.
Question 5.1
If u is parallel to v we say that u and v are linearly dependent, Why can linearly dependent vectors only form a one dimensional object? Explain with reference to the figure above.
5.2 Tiling the plane
Figure 5.2
Vectors u and v are given. A lattice is formed where a*u and b*v can be used to describe
w= a*u+b*v.
Note: u and v are initially set as linearly independent, where u is not parallel to v, and a 2D basis can be formed such that w= a*u+b*v.
5.2
Suppose you are given a new vector w that lies exactly on the line defined by u. In other words w is parallel to u. Can w still be described using the lattice w = au + bv?
5.3 - Orthogonal tiling
Figure 5.3 - an orthogonal tiling
figure shows vector u is perpendicular to vector u', we say these vectors are orthogonal.
We can use these vectors to tile the plane. Tiling can continue in all directions to infinity
5.4 - The Orthogonal tiling, where u is the unit vector at the origin.
we can now define the cartesian system
5.5 Define vectors in the plane
Question 5.5.1 - Define Vector PR in terms of vector u and v
Define Vector PR in terms of vector u and v
Question 5.5.2
Define Vector QS in terms of vector u and v
5.6 - Using Orthogonal Tiling to Define v and w.
5.6.1
Define W_2 by the vectors u and ⟂u
5.6.2
define v_2 by the vectors u and ⟂u
5.7 - Using the Cartesian System to define vectors.
5.7.1
Define vectors s, v and w by the orthogonal plane shown
5.7.2
using the values above, what is the sum of s-v