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Algebra for Athletes.com

8.3 Rotation Vectors

Up to this point, we've used vectors to show translation or movement from one point to another. Rotation can also be described using ijk unit vectors.  The unit rotation vectors describe the axis of rotation and its magnitude.  Since an object can rotate in two different directions on the same axis , defining the axis alone can't describe the rotation.  Rotation in one direction is considered positive while rotation in the other direction is considered negative.  To determine which is which, scientists and engineers use a principal called the right hand rule .  If the curved fingers of the right hand point in the direction of the rotation, the extended thumb will point in the positive direction of the axis.

The magnitude of the rotation vector is determined by the speed at which it's spinning or its angular velocity .  The magnitude of the angular velocity is given by the number of rotations per unit time.  The units of revolutions per minute or rpm's used with machinery is a unit of angular velocity.  Angular velocity is usually denoted with the Greek letter  (pronounced "omega") which is the lower case letter for "o".

The different types of pitches thrown in baseball help describe how rotation is defined.   For this description, we'll assume that all of the pitches have an angular velocity of a revolutions per second.  All of the pitches will be defined as being thrown by a right-handed pitcher away from the pitcher.

If we put the center of the ball at the origin of the xyz graph as shown in Fig. 8.11, we can describe its rotation by the ijk unit vectors.

The typical fastball is thrown with bottom spin and doesn't curve to the left or right.  It spins on a horizontal axis.  If you curve the fingers of your right hand in the direction of the spin, your extended thumb points in the positive direction of the y-axis.  Its rotation vector is therefore +aj.  A drop ball is given top spin.  It spins on the same axis as the fastball but in the opposite direction.  Its rotation vector is therefore -aj.  A slider is meant to curve to the left.  Its axis is vertical.  It has a rotational vector of +ak.[1] The screwball is meant to curve to the right, also spinning on a vertical axis but in the opposite direction of a slider. Its rotation vector is -ak.  The curve ball is a combination of a drop ball and a slider (preferably more drop than slider).  Its axis is tilted creating a rotation vector of about -bj + ck, where b²+ c²= a². What about the knuckle ball?  If thrown right, the knuckle ball  isn't given any spin.  Its rotation vector is 0i  + 0j + 0k.

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[1] Some players define a slider as also having a drop component like a curveball.