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402 Mastering Technical Mathematics
Figure 26-4
Two vectors a and
b in xyz-space.
They are added
using the parallelogram
method. This
is a perspective
drawing, so the
parallelogram
appears distorted.
DIRECTION IN XYZ SPACE
If you want a challenge, prove this fact on the basis of the two-dimensional Pythagorean
theorem for the magnitude of a vector a in xy space:
|a| (x a2
y a2
) 1/2
Now imagine that you want to find the magnitude of the vector denoted by a (x a
,y a
,z a
)
(1,2,3) in xyz space. Suppose you are told that the values 1, 2, and 3 are exact, and you
want to get the answer accurate to four digits after the decimal point. Use the distance
formula
|a| (x a2
y a2
z a2
) 1/2
(1 2 2 2 3 2 ) 1/2
(1 4 9) 1/2
14 1/2
3.7417
In three-dimensional xyz space, the direction of a vector a is denoted by measuring or
expressing the angles x
, y
, and z
that a subtends relative to the positive x, y, and z
axes, respectively, as shown in Fig. 26-5. These angles, given in radians as an ordered
triple ( x
, y
, z
), are called the direction angles of the vector a. So you can write
dir a ( x
, y
, z
)