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. [br][br]a, b are rational numbers

5.1 - Adding scaled vectors

Figure 5.1

Figure shows slider a and b[br]Vector u and v are given[br]the construction shows, vectors a*u and b*v head to tail [br]vector w = a*u + b*v[br][br]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, [br]Why can linearly dependent vectors only form a one dimensional object?[br]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 [br]w= a*u+b*v. [br]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.[br]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. [br]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, [br]what is the sum of s-v