Consider the rotation matrix in R given by Ro|cos@) sin( -sin( cos(9) Re are ort
ID: 3115781 • Letter: C
Question
Consider the rotation matrix in R given by Ro|cos@) sin( -sin( cos(9) Re are orthogonal. (Note, this means the columns form a basis for R -a (a) Show that the columns of useful property). (b) Show that the magnitude of each column is equal to 1. (Note, this means the columns vectors are unit vectors-also a useful property for a basis). (c) We can actually test parts (a) and (b) simultaneously using matrix multiplication using the transpose of a matrix: show that R3R . (Notice, 0's appearing off the main diagonal tell us what we found in part (a) -that the columns are orthogonal-and 1's appearing on the main diagonal tell us what we found in part (b) that the columns are unit vectors. We call a matrix satisfying ATA-1 an orthogonal matrix. These come in very handy for finding bases in real-world problemsExplanation / Answer
(a). The columns of R are u = (cos, sin)T and v = (-sin, cos)T. Further u.v = -sincos +-sincos = 0. Hence u and v are orthogonal.
(b) The magnitude of u , i.e. ||u|| = ( cos2+ sin2) = 1, as cos2+ sin2 = 1. Also, the magnitude of v , i.e. ||v|| = ( sin2+ cos2) = 1. Thus, the magnitude of each column of R is 1.
(c ) We have RT =
cos
sin
- sin
cos
Now the matrix multiplication is row by column so that RT R =
cos2 + sin2
-cos sin+ cos sin
- sin cos+ cos sin
sin2+ cos2
=
1
0
0
1
= I2
cos
sin
- sin
cos