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Similarly, β_x = γ_x and γ_y = α_y. We still need to figure out what the magnitudes of α_x, β_y, γ_z are. However, these terms are not used in our equations as derived above, so they can be set to any convenient value. Let us do the following:

Since β_x = γ_x, let α_x = β_x = γ_x

α_y = γ_y, let α_y = γ_y = β_y

α_z = β_z, let α_z = β_z = γ_z

If we can do this, then α = β = γ, and we can define an angular velocity of frame B with respect to frame A.

$\omega_{B/A} = \vec{\alpha} = \vec{\beta} = \vec{\gamma}$

and we see that

$\dot{\vec{i}}^{A}.\vec{j} = \alpha_{z} = \omega_{B/A}_{z}$

$\dot{\vec{j}}^{A}.\vec{k} = \beta_{x} = \omega_{B/A}_{x}$

$\dot{\vec{k}}^{A}.\vec{i} = \gamma_{y} = \omega_{B/A}_{y}$

Angular velocity of frame B with respect to frame A

$\dot{\vec{i}}^{A} + \dot{\vec{j}}^{A} + \dot{\vec{k}}^{A} = \omega_{B/A}\times\vec{i} + \omega_{B/A}\times\vec{j} + \omega_{B/A}\times\vec{k}$

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