Question

(f) [25 points] Quaternion and Rotation Properties 1. Recall that rotations can be composed by multiplying their quaternions, with the "most recent" rotation on the left. Use quaternion multiplication to show that rotations are not commutative. 2. Is quaternion multiplication associative? If no, provide a counterexample. If so, justify your reasoning briefly but do not prove it. 3. Suppose $\hat{q}$ is a normalized quaternion, representing a rotation. What rotation does $-\hat{q}$ represent? (HINT: rotating $\vec{v}$ by $\hat{q}$ results in $\hat{q}\vec{v}\hat{q}^{-1}$, and rotating it by $-\hat{q}$ results in $(-\hat{q})\vec{v}(-\hat{q}^{-1})$. 4. Imagine you rotate a vector by 90 degrees about the $\hat{x}$ axis, then 90 degrees about the $\hat{y}$ axis, and finally 90 degrees about the $\hat{z}$ axis. Use quaternion composition to show that this is equivalent to a single rotation of 90 degrees about the $\hat{y}$ axis. 5. A quaternion's real value is 1, and its directional components are all 0. Describe its encoded rotation.

          (f) [25 points] Quaternion and Rotation Properties
1. Recall that rotations can be composed by multiplying their quaternions, with the
"most recent" rotation on the left. Use quaternion multiplication to show that
rotations are not commutative.
2. Is quaternion multiplication associative? If no, provide a counterexample. If so,
justify your reasoning briefly but do not prove it.
3. Suppose $\hat{q}$ is a normalized quaternion, representing a rotation. What rotation
does $-\hat{q}$ represent? (HINT: rotating $\vec{v}$ by $\hat{q}$ results in $\hat{q}\vec{v}\hat{q}^{-1}$, and rotating it by $-\hat{q}$
results in $(-\hat{q})\vec{v}(-\hat{q}^{-1})$.
4. Imagine you rotate a vector by 90 degrees about the $\hat{x}$ axis, then 90 degrees about
the $\hat{y}$ axis, and finally 90 degrees about the $\hat{z}$ axis. Use quaternion composition to
show that this is equivalent to a single rotation of 90 degrees about the $\hat{y}$ axis.
5. A quaternion's real value is 1, and its directional components are all 0. Describe
its encoded rotation.

(f) [25 points] Quaternion and Rotation Properties
1. Recall that rotations can be composed by multiplying their quaternions, with the
"most recent" rotation on the left. Use quaternion multiplication to show that
rotations are not commutative.
2. Is quaternion multiplication associative? If no, provide a counterexample. If so,
justify your reasoning briefly but do not prove it.
3. Suppose q̂ is a normalized quaternion, representing a rotation. What rotation
does -q̂ represent? (HINT: rotating v⃗ by q̂ results in q̂v⃗q̂^-1, and rotating it by -q̂
results in (-q̂)v⃗(-q̂^-1).
4. Imagine you rotate a vector by 90 degrees about the x̂ axis, then 90 degrees about
the ŷ axis, and finally 90 degrees about the ẑ axis. Use quaternion composition to
show that this is equivalent to a single rotation of 90 degrees about the ŷ axis.
5. A quaternion's real value is 1, and its directional components are all 0. Describe
its encoded rotation.

Added by Laura O.

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