Rotating Vectors - Vector Analysis for Computer Graphics - page 194

Converting 2 q · v q :

2 q · v = 2xx v + yy v + zz v

Therefore,

2 q · v q = 2xx v + yy v + zz v x i + y j + z k

2 x 2 xz x v i 2 xy yz y v j 2 xz z 2 z v k

y 2 2 q · v q = + xy + + + + + + yz +

⎡ ⎤ ⎡ ⎤

2x 2 2xy 2xz
2xy 2y 2 2yz
2xz 2yz 2z 2

x v
y v
z v

⎣ ⎦ ⎣ ⎦

2 q · v q =

Converting 2s q × v :

i j k
xyz
x v y v z v

q × v = = yz v − zy v i + zx v − xz v j − xy v − yx v k

2s q × v = 2s yz v − zy v i + 2s zx v − xz v j − 2s xy v − yx v k

⎡ ⎤ ⎡ ⎤

0 − 2sz 2sy x v

y v
z v

⎣ ⎦ ⎣ ⎦

2s q × v = 2sz 0 − 2sx

− 2sy 2sx 0

Combining all three terms gives

⎡ ⎤ ⎡ ⎤

2s 2 2x 2 − 1 + 2xy − 2sz 2xz + 2sy x v

y v
z v

⎣ ⎦ ⎣ ⎦

2s 2 2y 2

qv ¯ q = 2xy + 2sz − 1 + 2yz − 2sx

2s 2 2z 2

2xz − 2sy 2yz + 2sx − 1 +

Simplifying leads to

⎡ ⎤ ⎡ ⎤

2s 2

+

x 2

− − +

12 xy sz 2xz sy x v

y v
z v

⎣ ⎦ ⎣ ⎦

+

2s 2

+

y 2

− −

qv q ¯ = 2xy sz 12 yz sx

2xz − sy 2yz + sx 2s 2 + z 2 − 1

or

⎡ ⎤ ⎡ ⎤

1 − 2y 2 + z 2 2xy − sz 2xz + sy x v

y v
z v

⎣ ⎦ ⎣ ⎦

¯ = 2xy + sz 1 − 2x 2 + z 2 2yz − sx

qv q

2xz − sy 2yz + sx 1 − 2x 2 + y 2

This is an important matrix as it is used to represent quaternion operations within computer
graphics programs.
It will be interesting to see the matrix version of Example 2 above and show that it produces
an identical result. To recap, the angle is 180 , the axis is

1
√ 2 − i + j