## Addition and Subtraction of Vectors

Mechanics of human movement can best be explained with the use of vector notation and calculus. We present next a brief review of vector mathematics. A vector is a quantity that has both a magnitude and direction. Perhaps the vector most commonly known is body weight, which acts always in the direction pointing to the center of earth. Graphically, a vector is shown as a directed segment of a straight line. The length of the segment represents the magnitude of the vector. The direction of the line segment is the direction of the vector. The sense of direction is from the tail end of the segment to the end capped by an arrow. In this text we use boldface type when referring to a vector. As illustrated in Fig. 2.3, vectors are added by bringing them together arrow to tail and then by connecting the free tail with that of the free arrow. This is called the parallelogram law. The parallelogram law can be used to evaluate the re- Figure 2.3a,b. The parallelogram law of addition of two vectors. Two vectors are added by bringing them together arrow to tail and then by connecting the free tail with that of the free arrow (a). The application of this so-called parallelogram law to the calf muscle is shown in (b).

Figure 2.3a,b. The parallelogram law of addition of two vectors. Two vectors are added by bringing them together arrow to tail and then by connecting the free tail with that of the free arrow (a). The application of this so-called parallelogram law to the calf muscle is shown in (b).

sultant force acting on a joint or a limb. For example, as shown in Fig. 2.3b, the resultant force exerted by the calf muscle on the ankle is the sum of the muscle force produced by the two fleshy bellies of this muscle. This resultant acts as the propelling force during jumping and sprinting.

Both the commutative law and the associative law hold for vector addition:

Vector subtraction can be thought of as a special case of addition:

in which — b is equal to vector b in magnitude but has the opposite sense of direction. Thus vector subtraction has all the properties associated with vector addition.

Vectors can also be added algebraically. For this purpose, one draws three mutually perpendicular straight lines protruding from a point O in space. This is called a Cartesian reference frame. The point O is termed the origin of the coordinate system and the three straight lines stemming from it are the coordinate axes. Let E (x\, x2, x3) be a Cartesian coordinate system and let e1, e2, and e3 denote unit vectors along x1, x2, and x3 directions as shown in Fig. 2.4. A unit vector is defined as a vector with a unit magnitude. Any vector b can be expressed in this coordinate system as the sum of three components:

in which the symbols b2, and b3 represent the projections of vector b on xi, x2, and x3 axes. These projections are visualized in Fig. 2.4. To determine projection b1, one draws perpendicular lines from each end of the  Figure 2.4a,b. Representation of a vector in a Cartesian reference frame E. The three mutually perpendicular lines protruding from point O compose the reference frame E. The symbols ei, e2, and e3 denote unit vectors along the coordinate axes. The symbols b1, b2, and b3 denote, respectively, the projections of b onto the directions specified by the unit vectors ej, e2, and e3. Shown in (a) is a vector originating from the origin of the reference frame. Fig. 2.6b illustrates a vector whose origin does not coincide with the origin of the reference frame. The parameter bj is negative if the projection is in the opposite direction of the unit vector ei.

Figure 2.4a,b. Representation of a vector in a Cartesian reference frame E. The three mutually perpendicular lines protruding from point O compose the reference frame E. The symbols ei, e2, and e3 denote unit vectors along the coordinate axes. The symbols b1, b2, and b3 denote, respectively, the projections of b onto the directions specified by the unit vectors ej, e2, and e3. Shown in (a) is a vector originating from the origin of the reference frame. Fig. 2.6b illustrates a vector whose origin does not coincide with the origin of the reference frame. The parameter bj is negative if the projection is in the opposite direction of the unit vector ei.

vector b to the x1 axis. The line segment that remains between the two intersections represents b1. Projections b2 and b3 are determined similarly. A projection can be positive or negative. A projection is positive if it points along one of the unit vectors ei, e2, and e3; otherwise, it is negative. Note that in most undergraduate texts on dynamics, the vectors ej, e2, and e3

are represented by the symbols i, j, and k. The notation adopted in this text is especially useful when considering multibody systems such as the human body.

The magnitude of vector b is denoted as ||b||, and by Pythagoras' theorem it is equal to

The unit vector along the direction of b can be found by dividing b by its magnitude eb = b/||b|| (2.2c)

Note that a vector can be divided by a scalar by dividing its projections along the coordinate axes with that scalar. Division of a vector by another vector is not an operation that has been defined. So, a vector cannot be divided by another vector.

Two vectors a and b are equal to each other if and only if their three components along the axes of a reference frame are equal to each other:

a = b if and only if a1 = b1; a2 = b2; a3 = b3 (2.3)

In a Cartesian coordinate system, two vectors a and b are added algebraically as follows:

a + b = (a1 + b1) e1 + (a2 + b2) e2 + (a3 + b3) e3 (2.4)

A vector is multiplied by a positive number (a scalar) when its magnitude is multiplied by that number. If a vector is multiplied by a negative number, its direction is reversed and its magnitude multiplied by the absolute value of the number. The following equations hold for multiplication of vectors by a scalar:

Example 2.1. Elbow Force During Baseball Pitching. Baseball players, especially pitchers, are prone to overuse injuries associated with throwing. These injuries result from accumulated microtrauma developed during repetitive use. To test the hypothesis that the microtrauma in the throwing arm of a pitcher is caused by the large forces and torques exerted at the shoulder and elbow joint during pitching, a large number of studies have investigated the mechanics of pitching using high-speed motion analysis. The forces applied by the ligaments and tendons on the elbow joint during baseball pitching were measured in the medial (M), anterior (A), and compression (C) directions (Fig. 2.5). The magnitudes of these forces were found to be

Figure 2.5. Forces exerted at the elbow by the rest of the body in the medial (M), anterior (A), and compressive (C) directions during pitching. These forces have been estimated from the three-dimensional videoanalysis of a pitching event using the method of inverse dynamics. where N denotes the force unit Newton. A force that accelerates 1 kg at a rate of 1 m/s2 is 1 N.

The unit vectors in the medial, anterior, and compression directions were expressed in unit vectors fixed on earth (e1, e2, e3,):

Using these data, compute the resultant force acting on the elbow.

Solution: The resultant force acting on the elbow is equal to the sum of the three forces acting on the elbow:

in which F stands for the resultant force.

Substituting the expressions given for the magnitude of the forces and their directions into this expression, we obtain:

F = 428 (0.79 e1 + 0.17 e2 + 0.59 e3) + 101 (0.21 e1 - 0.98 e2)

Summing the coefficients in front of the unit vectors, this expression can be put into the following simpler form:

We can express the resultant force as the product of its magnitude times the unit vector in the direction of the resultant force:

The magnitude of the resultant force was determined by using Eqn. 2.2b. The unit vector along the direction of force was obtained by dividing the resultant force by its magnitude.

Example 2.2. Resultant Force Exerted by the Pectoralis on the Upper Arm. The man shown in Fig. 2.6 is performing lateral flies to work his pectoralis muscles. The pectoralis is a triangular muscle of the upper chest; Figure 2.6a,b. A man exercising his pectoralis muscle group (a). The schematic diagram of the pectoralis as composed of three distinct sets of muscle fibers (b).

assume that it can be represented as composed of three sets of muscle fiber groups connecting the sternum to the humerus. Determine the resultant force exerted by the pectoralis on the humerus. At the position shown in the figure, the muscle acts in a plane, and the magnitude of the force produced by each set of fibers is 75 N.

Solution: The resultant pectoralis force F is the sum of the forces produced by the three sets of fibers

||F|| = [3 + 2(cos 10° + cos 20° + cos 30°)]1/2 75 N

In deriving the equation for the magnitude of the pectoralis force, we used the trigonometric relation cos (a - b) = cos (a) cos (b) + sin (a) sin (b)

In this example, we have assumed that all fibers of the pectoralis were activated by the central nervous system. The pectoralis is capable of exerting forces on the humerus in wide-ranging directions. This force is accomplished by varying the spatial activation pattern of the muscle itself. 