my question comes from the portion of the derivation regarding evaluating the rate of change of the principle axis vectors. this begins by supposing a vector, Q, is rotating about axis n by δθ. Specifically, my question is how from step 4 to step 5 the approximation becomes an equality.
Q' can be approximated as:
1. Q' ≈ Q + (|Q|sinα)δθ(in direction of n cross Q)
2. Q' ≈ Q + |n cross Q|(in direction of n cross Q)δθ
3. Q' ≈ Q + (n cross Q)δθ
4. Q' ≈ Q + δθ cross Q
The derivation then states: "This means that we can express the derivative of this vector as:
5. (Q'-Q)/dt = (δθ cross Q)/dt
6. dQ/dt = δθ/dt cross Q
7. dQ/dt = Ω cross Q
Q' can be approximated as:
1. Q' ≈ Q + (|Q|sinα)δθ(in direction of n cross Q)
2. Q' ≈ Q + |n cross Q|(in direction of n cross Q)δθ
3. Q' ≈ Q + (n cross Q)δθ
4. Q' ≈ Q + δθ cross Q
The derivation then states: "This means that we can express the derivative of this vector as:
5. (Q'-Q)/dt = (δθ cross Q)/dt
6. dQ/dt = δθ/dt cross Q
7. dQ/dt = Ω cross Q