Scott McClelland

r-mcclelland

Assignment 1

Problem 1.3.3

Restatement:

        r₁ = 2i + j - 3k    r₂ = i + j + k

Redefine coordinates in R³ by:

        x' = x -1    y' = y + 2    z' = z

Calculate the primed coordinates of the vectors r₁ and r₂, and

 verify that r₂ - r₁ = r₂' - r₁' and ||r₂ - r₁|| = ||r₂' - r₁'||.

Solution:

First, we will find the primed coordinates of the vectors by taking the old x_i value and

subtracting 1 from it.  Similarly, we would add 2 to the old y_j coordinate and then leave

the z_k coordinate the same.  This will give us the new vectors:

        r₁' = i + 3j - 3k    r₂' = 3j + k

Then, we will verify:

r₂-r₁ = <1,1,1> - <2,1,-3> = <-1,0,4>

r₂'-r₁' = <0,3,1> - <1,3,-3> = <-1,0,4>

||r₂-r₁|| = √(x_i² + y_j² + z_k²) = √((-1)² + 0² + (4)²) = √(17)

||r₂'-r₁'|| = √(x_i² + y_j² + z_k²) = √((-1)² + 0² + (4)²) = √(17)