Introduction to Sports Biomechanics: Analysing Human Movement ...

INTRODUCTION TO SPORTS BIOMECHANICS
Vector multiplication
The rules for vector multiplication do not follow the simple algebraic rules for multiplying
scalars.
Vector (cross) product
Figure 4.24 Vector cross-product.
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The vector (or cross) product of two vectors is useful in rotational motion because it
enables, for example, angular motion vectors to be related to translational motion
vectors (see below). It will be stated here in its simplest case for two vectors at right
angles, as in Figure 4.24. The vector product of two vectors p and q inclined to one
another at right angles is defined as a vector p × q of magnitude equal to the product
(p q) of the magnitudes of the two given vectors. Its direction is perpendicular to both
vectors p and q in the direction in which the thumb points if the curled fingers of the
right hand point from p to q through the right angle between them.
Scalar (dot) product
The scalar (dot) product of two vectors can be used, for example, to calculate power
(a scalar, P) from force (F) and velocity (v), which are both vectors, using P = F.v. The
dot product is a scalar so has no directional property. It is calculated as the product of
the magnitudes of the two vectors and the cosine of the angle between them. It can also
be calculated from the components of the vectors. For example, if F = F x + F y and v = v x
+ v y then P = (F x + F y).(v x + v y). Now, the angle between the force and velocity along the
same axis is 0°, so the cosine of that angle is 1; however, the angle between the force and
velocity along different axes is 90°, so the cosine of that angle is 0. Hence, P = F xv x +
F yv y. Please note that the power does NOT have x and y components, as it is a scalar (see
also Figure 5.25(i)).