Vectors

A "vector" is one of those mathematical terms that make most people's eyes instantly glaze over. For a martial artist, though, the concept of a vector is quite useful to know. A vector is defined as:

Vectors are important to martial artists because it allows us to understand and use the principles of momentum, which is a measurement of an object's kinetic energy or energy when an object is in motion relative to another object. Any force is actually a vector, having a direction and a magnitude. Any strike can be conceptualized as a vector. While it is clear that a direct hit will cause maximum damage, it isn't quite as clear that the direction of the attack is also important. By examining vectors, it is easier to see what really happens when a target gets hit. Figure 1 shows two different punches. The bottom punch is directly toward the target. F is the force that's exerted on the board when the fist slams into it. For the punch on the top, only a part of the force F is directed toward the board.

Figure 1 - Vectors for punches

The actual force of a strike is somewhat more complicated, but we'll simply use Newton's Third Law and assume that the force that the board will feel is the force that was required to accelerate the punch. To see how much power is delivered to the target, we can use the components of F, a, and b as seen in the top of Figure 1. Some of this force will be directed on the board (although the same is true of a martial artist's hand as well, according to Newton's third law). The component b doesn't contribute to the break. The component vector (to use the math-ese) a is defined as straight toward the board, and b as perpendicular to a, so F, a, and b form a right triangle (where a and b are 90° apart). We can use trigonometry to find a.

Equation 1:

Θ(Theta) = force, a = force toward the board, F = Force (hypotenuse)

If F = 1000 N, then the force a will be (cos 20°) * 1000, or 939.69, or 94% of the total force. Just by hitting the target perpendicular, the martial artist gains 60 N (i.e. by hitting it at the wrong angle, the martial artist lost 6% of her power.) Hitting targets directly may seem obvious, but it isn't quite as obvious in practice.