082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

8.4 Using matrices in the translation and rotation of vectors 269
Itisthen possible todefine several matrices:
⎛
⎞
1 0 0 0
Rot(x,θ) = ⎜0 cosθ −sinθ 0
⎟
⎝0 sinθ cosθ 0⎠ (8.2)
0 0 0 1
⎛
⎞
cosθ 0 sinθ 0
Rot(y,θ) = ⎜ 0 1 0 0
⎟
⎝−sinθ 0 cosθ 0⎠ (8.3)
0 0 0 1
⎛
⎞
cosθ −sinθ 0 0
Rot(z,θ) = ⎜sinθ cosθ 0 0
⎟
⎝ 0 0 1 0⎠ (8.4)
0 0 0 1
⎛ ⎞
100a
Trans(a,b,c) = ⎜010b
⎟
⎝001c⎠ (8.5)
0001
Matrices (8.2) to (8.4) allow vectors to be rotated by an angle θ around axesx,yandz,
respectively. For example, the product Rot(x,θ)V has the effect of rotating r through
anangle θ about thexaxis. Matrix (8.5)allows a vector tobetranslatedaunits inthex
direction,bunits intheydirection andcunits inthezdirection.
It is possible to combine these matrices to calculate the effect of several operations
on a vector. In doing so, it is important to maintain the correct order of operations as
matrix multiplication isnon-commutative.
For example, the position of a vector that has first been translated and then rotated
about thexaxis can be defined by
V new
= Rot(x,θ)Trans(a,b,c)V old
Afewexamples will help toclarifythese ideas.
Example8.12 Rotate the vector
⎛ ⎞
1
r = ⎝1⎠
2
through 90 ◦ about thexaxis.
⎛ ⎞
⎛ ⎞ 1
1
Solution r old
= ⎝1⎠ V old
= ⎜1
⎟
⎝2⎠
2
1
Rot(x,90 ◦ ) =
⎛
⎞ ⎛ ⎞
1 0 0 0 10 00
⎜0 cos90 ◦ −sin90 ◦ 0
⎟
⎝0 sin90 ◦ cos90 ◦ 0⎠ = ⎜00−10
⎟
⎝01 00⎠
0 0 0 1 00 01