Page 40 - Dynamics of Mechanical Systems

0593_C02_fm Page 21 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 21

Then, the magnitude of A is simply:

/
A = a 2 + a 2 y + a 2 z ) 12 (2.4.10)
( x
Next, suppose that n is a unit vector parallel to A, as in Figure 2.4.3. Then, by following
the same procedure as in Eq. (2.4.9), n may be expressed as:

n = cosθ n + cosθ n + cosθ n (2.4.11)
x x y y z z
Because the magnitude of n is 1, we then have:

1 = cos θ + cos θ + cos θ (2.4.12)
2
2
2
x y z
Example 2.4.1: Vector Addition Using Components
To illustrate the use of these ideas, consider again the vectors of Examples 2.3.1 and 2.3.2
(see Figures 2.3.6 and 2.3.7). Specifically, let A be parallel to the Y-axis with a magnitude
of 15 N. Let B be parallel to the Y–Z plane and inclined at 60˚ relative to the Y-axis. Let
the magnitude of B be 12 N. Let C be parallel to the X-axis with a magnitude of 10 N
(Figure 2.4.4). As before let n , n , and n be unit vectors parallel to the X-, Y-, and Z-axes.
x
z
y
Then, A, B, and C may be expressed as:
A = 15 n N
x
.
B = 12cos 60 ° n + 12sin 60 ° n = 6 n + 10 392 n N (2.4.13)
y z y z
C = 10 n N
x
Hence, R becomes:

)
(
R = A B C = 15 n + 6 n + 10 392 n + 10 n N
+
+
.
y y z x
(2.4.14)
.
= 10 n + 21 n + 10 392 n N
x y z
Then, the magnitude of R is:
[ 2 2 ) 2 ] 12
/
21
.
10
.
R = () +() +(10 392 N = 25 47N (2.4.15)
This is the same result as in Example 2.3.2 (see Eq. (2.3.8)). Here, however, we obtained
the result without using the law of cosines as in Eq. (2.3.4).
Example 2.4.2: Direction Cosines
A particle P is observed to move on a curve C in a Cartesian reference frame R, as shown
in Figure 2.4.5. The coordinates of P are functions of time t. Suppose that at an instant of
interest x, y, and z have the values 8 m, 12 m, and 7 m, respectively. Determine the
orientation angles of the line of sight of an observer of P if the observer is at the origin
O. Specifically, determine the angles θ , θ , and θ of OP with the X-, Y-, and Z-axes.
x
z
y