Graphics Reference
In-Depth Information
2.13 Direction cosines
A vector's orientation and magnitude are encoded within its Cartesian components. The
magnitude of a 3D vector v is readily obtained using
x v +
v
=
y v +
z v
while its orientation is obtained using
tan 1 y v
x v
=
where is the angle between the vector and the x-axis. However, we can develop this idea of
orientation using direction cosines .
Y
Y
v
v
α
j
α
α j
α k
i
α
i
X
X
Z
Figure 2.40.
Figure 2.40 illustrates the idea behind direction cosines for a 2D and a 3D vector. The angles
i , j , and k (for a 3D vector) provide an elegant visual mechanism for orientating a vector
relative to the Cartesian axes. In fact, the cosines of the angles are used as they are intimately
related to the vector's components.
Y
y v
v
α j
α i
x v
X
Figure 2.41.
To begin with, consider the 2D vector v in Fig. 2.41. It has components x v and y v and it is
obvious that i +
90 . Now
j =
x v
cos i =
v
 
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