Leaving room for implied movement of your subject is important in creating a composition that doesn’t express tension. This also applies to subjects that are looking to the left or right, up or down. If the subject’s eyes are pointed to the right and the subject is placed on the right side of the frame, an uneasy balance is created as the image is too heavily weighted to that area of the photograph.
To avoid compositional tension or over-weighting of one side, frame the image so the subject is placed on the opposite side from the direction of movement or direction of stare. Frame the image so the assumed motion of the subject doesn’t collide with the edge of the photograph.
The shot of the red tail hawk in flight works because the movement of the bird is from right to left and the hawk is positioned on the right side of the frame. If the hawk was centered, it would still work, but not as successfully. On the other hand, if the hawk appeared on the left side of the image, it would fall short as the bird would appear to be crashing into the edge of the frame. Additionally, the entire right side would have no areas of interest.