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In this video well see how the matrix exponential can be applied to integrate the angular velocity of a rotating rigid body. Lets start with this frame of coordinate axes, which will rotate about a unit angular velocity axis. To understand the motion of the coordinate axes, it suffices to consider just one of the coordinate axes, since the same reasoning applies to any axis. Lets call this remaining vector p. As the vector p rotates about the rotation axis, it traces out a circle. The purpose of this video is to determine the final location of the vector if it rotates an angle theta about the rotation axis. We will do this by integrating the differential equation of motion describing the motion of p. Here is a picture of our initial vector, p at time 0, and the unit rotation axis omega-hat. As p begins to rotate, it traces out a circle around the rotation axis. The 3-vector linear velocity is tangent to the circle at any time, and is given by omega-hat cross p. After rotating an an