1. Let S2 {B1, B2} be an ordered basis set of vectors in 2-D. (a) Let ß1 = i, B2 = j in Cartesian coordinates on the xy-plane. Find the 2 × 2 matrix R1 that represents, on basis S2, a clockwise ro- tation by angle 30° in the plane, by using trigonometry to determine explicitly how this rotation would transform each basis %3D vector. (b) Now, define R2 to be the 3 x 3 matrix in 3-D which represents basis S3 on an expanded {B1, B2, B3} with B3 = k - a - clockwise rotation by angle 30° about the positive z-axis. Noting that k doesn't ro- tate, extend the result in part (a) to 3-D to find R2. (c) Taking inspiration from parts (a) and (b), find R3 in the same basis S3 for 30° rota- tion about x-axis clockwise. (d) Similar to (c), find R4 in the same basis S3 for 30° rotation about y-axis clockwise.
1. Let S2 {B1, B2} be an ordered basis set of vectors in 2-D. (a) Let ß1 = i, B2 = j in Cartesian coordinates on the xy-plane. Find the 2 × 2 matrix R1 that represents, on basis S2, a clockwise ro- tation by angle 30° in the plane, by using trigonometry to determine explicitly how this rotation would transform each basis %3D vector. (b) Now, define R2 to be the 3 x 3 matrix in 3-D which represents basis S3 on an expanded {B1, B2, B3} with B3 = k - a - clockwise rotation by angle 30° about the positive z-axis. Noting that k doesn't ro- tate, extend the result in part (a) to 3-D to find R2. (c) Taking inspiration from parts (a) and (b), find R3 in the same basis S3 for 30° rota- tion about x-axis clockwise. (d) Similar to (c), find R4 in the same basis S3 for 30° rotation about y-axis clockwise.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Transcribed Image Text:1. Let S2
{B1, B2} be an ordered basis set of
vectors in 2-D.
(a) Let B1 = î, B2 = j in Cartesian coordinates
on the xy-plane. Find the 2 × 2 matrix R1
that represents, on basis S2, a clockwise ro-
tation by angle 30° in the plane, by using
trigonometry to determine explicitly how
this rotation would transform each basis
vector.
(b) Now, define R2 to be the 3 x 3 matrix in
3-D which represents
basis S3 = {B1, B2, B3} with B3
clockwise rotation by angle 30° about the
positive z-axis. Noting that k doesn't ro-
tate, extend the result in part (a) to 3-D
to find R2.
on an expanded
-
a
(c) Taking inspiration from parts (a) and (b),
find R3 in the same basis S3 for 30° rota-
tion about x-axis clockwise.
(d) Similar to (c), find R4 in the same basis S3
for 30° rotation about y-axis clockwise.
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