Question

(4) (a) Determine the standard matrix A for the rotation r of R3 around the z-axis Hint: Use the matrix for the rotation around the origin in R2 for the ry-plane. (b) Consider the rotation s of R around the line spanned by El through the angle through the angle π/3 counterclockwise. onsider the rotation s of lR around the Iine spaned by /3 counterclockwise. Find a basis of R3 for which the matrix [s]в.в is equal to A from (a). (c) Give the standard matrix [SEE for the standard basis E (You do not need to actually multiply and invert the involved matrices; the product formula is enough

(4) (a) Determine the standard matrix A for the rotation r of R 3 around the z-axis through the angle π/3 counterclockwise. Hint: Use the matrix for the rotation around the origin in R 2 for the xy-plane. (b) Consider the rotation s of R 3 around the line spanned by h 1 2 3 i through the angle π/3 counterclockwise. Find a basis of R 3 for which the matrix [s]B,B is equal to A from (a). (c) Give the standard matrix [s]E,E for the standard basis E (You do not need to actually multiply and invert the involved matrices; the product formula is enough).

Answer 1

$\large \\ \textbf{(a)} \\ Consider \ the \ general \ formula \ as \ follows: \\ If \ u=(a,b,c) \ is \ a \ unit \ vector, \ then \ the \ standard \ matrix \\ R_{u, \theta} \ for \ the \ rotation \ through \ the \ angle \ \theta \ about \ an \ axis \\ through \ the \ origin \ with \ orientation \ u \ is \\ R_{u, \theta}= \begin{bmatrix} a^2(1-\cos \theta) +\cos \theta & ab(1-\cos \theta)-c \sin \theta & ac (1-\cos \theta)+b \sin \theta \\ ab (1-\cos \theta)+c \sin \theta & b^2 (1-\cos \theta)+\cos \theta & bc (1-\cos \theta)-a \sin \theta \\ ac (1-\cos \theta)-b \sin \theta & bc (1-\cos \theta) +a \sin \theta & c^2(1-\cos \theta)+\cos \theta \end{bmatrix} \\ In \ our \ given \ problem, \ the \ axis \ of \ rotation \ is \ z-axis \\ i.e., \ u=(0,0,1) \ and \ \theta=\frac{\pi}{3}. \ Thus \ the \ rotation \ matrix \ is \ given \ by \\ R_{u, \frac{\pi}{3}}= \begin{bmatrix} \frac{1}{2} & -\frac{\sqrt 3}{2} & 0 \\ \frac{\sqrt 3}{2} & \frac{1}{2} & 0 \\ 0 & 0 & 1 \end{bmatrix} \\$