Rotation Matrices - University of Utah

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Rotation Matrices

4.

Suppose that ? 2 R. We let

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R ? : R 2 ! R2

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be the function defined as follows:

Any vector in the plane can be written in polar coordinates as r(cos(?), sin(?))

where r 0 and ? 2 R. For any such vector, we define

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c)

R? r( cos(?), sin(?) ) = r( cos(? + ?), sin(? + ?) )

c)

Notice that the function R? doesn¡¯t change the norms of vectors (the number r), it just a?ects their direction, which is measured by the unit circle

coordinate.

We call the function R? rotation of the plane by angle ?.

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c)

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c)

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If ? > 0, then R? rotates the plane counterclockwise by an angle of ?. If

? < 0, then R? is a clockwise rotation by an angle of |?|. The rotation does

not a?ect the origin in the plane. That is, R? (0, 0) = (0, 0) always, no matter

which number ? is.

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Examples.

? R ¦Ð2 is the function that rotates the plane by an angle of ¦Ð2 , or 90? .

Because ¦Ð2 > 0, it is a counterclockwise rotation. Thus, R ¦Ð2 (1, 1) is the point

in the plane that we obtain by rotating (1, 1) counterclockwise by an angle

of ¦Ð2 .

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¡ª

(H)

a

¡®

S

? Because ? ¦Ð2 < 0, R? ¦Ð2 is a clockwise rotation. R? ¦Ð2 (1, 1) is the point

in the plane obtained by rotating (1, 1) clockwise by an angle of ¦Ð2 .

(I,¡¯)

? The function R0 : R2 ¡ú R2 rotates the plane by an angle of 0. That

is, it doesn¡¯t rotate the plane at all. It¡¯s just the identity function for the

plane.

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? Below is the picture of a shape in the plane. It¡¯s a triangle, and we¡¯ll

call this subset of the plane D.

D

R ¦Ð2 (D) is the set in the plane obtained by rotating D counterclockwise by

an angle of ¦Ð2 . (It¡¯s counterclockwise because ¦Ð2 > 0.)

R? ¦Ð4 (D) is D rotated clockwise by an angle of ¦Ð4 .

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? Let¡¯s rotate the vector (a, 0), where a ¡Ý 0. This is a point on the

x-axis whose norm equals a.

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(ao¡¯):o.(l,o)

(i,o)

We can write this vector in polar coordinates as a(1, 0), or equivalently, as

a( cos(0), sin(0) ). Now we can rotate the vector (a, 0) by an angle ¦Á. That¡¯s

the vector R¦Á (a, 0), which by the formula from the beginning of this chapter

is




R¦Á (a, 0) = R¦Á a( cos(0), sin(0) )

(o¡¯i)v:

(o¡¯i)v:

= a( cos(0 + ¦Á), sin(0 + ¦Á) )

(o¡¯i)v:

= a( cos(¦Á), sin(¦Á) )

(o¡¯i)v:

(o¡¯z) ¡®¡ã

(o¡¯z) ¡®¡ã

(o¡¯z) ¡®¡ã

0

(o¡¯z) ¡®¡ã

0

(o¡¯i)v:

? In this example, we¡¯ll rotate a vector (0, b), where b ¡Ý 0. This is a

vector whose norm


equals


b, and that points straight up. In polar coordinates,

¦Ð

¦Ð

(0, b) = b( cos 2 , sin 2 ).

(o¡¯z) ¡®¡ã

(o,)

fir

¡°a

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Now if we rotate (0, b) by an angle ¦Á, then we have

 

¦Ð 

 ¦Ð  

R¦Á (0, b) = R¦Á b cos

, sin

2



¦Ð



 ¦Ð2



= b cos

+ ¦Á , sin

+¦Á

2

2







¦Ð 

¦Ð

, sin ¦Á +

= b cos ¦Á +

2

2

There¡¯s a slightly better way to write the result above, but it requires a couple of the identities we learned in the chapter ¡°Sine and Cosine¡±. Specifically,

Lemmas 8-10 tell us that



¦Ð

¦Ð

= cos ¦Á + ¦Ð ?

cos ¦Á +

2

2

= sin(¦Á + ¦Ð)



[Lemma 9]

= ? sin(¦Á)

[Lemma 10]

and

¦Ð

sin ¦Á +

= cos(¦Á)

[Lemma 8]

2

Therefore,







¦Ð 

¦Ð

, sin ¦Á +

= b( ? sin(¦Á), cos(¦Á) )

R¦Á (0, b) = b cos ¦Á +

2

2



(o,)

cos (oc))

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