Introduction to Sports Biomechanics: Analysing Human Movement ...

INTRODUCTION TO SPORTS BIOMECHANICS
vector composition and can be achieved using a vector parallelogram or vector triangle.
This is illustrated by the examples of Figure 4.22, where the blue arrows are the original
vector and the black ones are its components. The vector parallelogram approach is
more usual as the components then have a common origin, as in Figure 4.22(a).
Most angular motion vectors obey the rules of resolution and composition, but this
is not true for angular displacements.
Figure 4.22 Vector resolution using: (a) vector parallelogram; (b) vector triangle.
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Vector addition and subtraction using vector components
Vector addition and subtraction can also be performed on the components of the vector
using the rules of simple trigonometry. For example, consider the addition of the three
vectors represented in Figure 4.23(a). Vector A is a horizontal vector with a magnitude
(proportional to its length) of 1 unit. Vector B is a vertical vector of magnitude −2 units
(it points vertically downwards, hence it is negative). Vector C has a magnitude of
3 units and is 120° measured anticlockwise from a right-facing horizontal line.
The components of the three vectors are summarised below (see also Figure 4.23(b)).
The components (Rx, Ry) of the resultant vector R = A + B + C are shown in
Figure 4.23(c). The magnitude of the resultant is then obtained from the magnitudes of
2 1/2
), as R = (0.25 + 0.36)
its two components, using Pythagoras’ theorem (R 2 = R x
2 + Ry
= 0.78. Its direction to the right horizontal is given by the angle θ, whose tangent is
R y/R x. That is tan θ = −1.2, giving θ = 130°. The resultant R is rotated anticlockwise
130° from the right horizontal, as shown in Figure 4.25(d). (A second solution for tan θ
= −1.2 is θ = −50°, which would have been the answer if R x had been + 0.5 and R y had
been −0.6).
Vector Horizontal component (x) Vertical component (y)
A 1 0
B 0 −2
C 3 cos120° = −3 cos60° = −1.5 3 sin120° = 3 cos60° = 2.6
A + B + C −0.5 0.6