平面向量的旋转
10-06
数值分析
1.140页,旋转变换
(1)关于平面向量的逆时针旋转:
已知任意一个向量OA=(x,y),把向量OA 绕其起点O 沿逆时针方向旋转α角得到向量OB=(xcosα-ysin α,xsia α+ycosα) 。推导如下:
OA =OB =x 2+y 2
sin θ=
y x +y
2
2
,cos θ=
x x +y
2
2
sin(α+θ) =
q x +y
2
2
=sin α⋅cos θ+sin θ⋅cos α=
x ⋅sin αx +y
2
2
+
y ⋅cos αx +y
2
2
所以:q =x ⋅sin α+y ⋅cos α
cos(α+θ) =
p x +y
2
2
=cos α⋅cos θ-sin α⋅sin θ=
x ⋅cos αx +y
2
2
-
y ⋅sin αx +y
2
2
所以:p =x ⋅cos α-y ⋅sin α
(2)关于平面向量的顺时针旋转:
已知任意一个向量OB=(p,q),把向量OB 绕其起点O 沿顺时针方向旋转α角得到向量OA=(q ⋅sin α-p ⋅cos α, p ⋅sin α+q ⋅cos α) 。推导如下:
OA =OB =
p 2+q 2
sin θ=
q p +q
2
2
,cos θ=
-p p +q
2
2
sin(α+θ) =
y p +q
2
2
=sin α⋅cos θ+sin θ⋅cos α=
-p ⋅sin αp +q
2
2
+
q ⋅cos αp +q
2
2
所以:y =-p ⋅sin α+q ⋅cos α
cos(α+θ) =
-x p +q
2
2
=cos α⋅cos θ-sin α⋅sin θ=
-p ⋅cos αp +q
2
2
-
q ⋅sin αp +q
2
2
所以:x =p ⋅cos α+q ⋅sin α