Page 47 - Dynamics of Mechanical Systems

0593_C02_fm Page 28 Monday, May 6, 2002 1:46 PM

28 Dynamics of Mechanical Systems

Hence,

A = 45 (2.6.29)

Example 2.6.3: Angle Between Two Vectors
Let vectors A and B be expressed in terms of unit vectors n , n , and n as:
2
3
1
−
A = 3 n n + 4 n and B = −5 n + 2 n + 7 n (2.6.30)
2 3 1 2 3
Determine the angle between A and B.
Solution: From Eq. (2.6.25), cosθ is:
⋅
3
5
4
1 2
cosθ= AB = () − ( ) +− ( )( ) + ()( ) 7 = . 0 244 (2.6.31)
AB 26 78
Then, θ is:
θ= cos −1 ( .0 244 ) = 75 .86 deg (2.6.32)

Example 2.6.4: Projection of a Vector Along a Line
Consider a velocity vector V and points P and Q of a Cartesian coordinate system R as
shown in Figure 2.6.6. Let V be expressed in terms of mutually perpendicular unit vectors
i, j, and k as:

V = 3 i + 4j + 4 kft sec (2.6.33)
Let the coordinates of P and Q be as shown and let L be a line passing through P and Q.
Determine the projection of V along L.
Solution: From Eq. (2.6.9), the projection of V along L is simply where n is a unit vector
along L. Because L passes through P and Q, n is:

i (
n = PQ PQ = − + 5 j− 2 k) 29 (2.6.34)
Therefore, the projection of V along (or onto) L is:

i (
⋅
vn = (3 i + 4 j + 4 k) ⋅− + 5 j − 2 k) 29
(2.6.35)
= () − ( ) + ()( ) + () − ( )] 29 = 1 67 ft sec
2
.
4
1
[ 3
4 5
2.7 Vector Multiplication: Vector Product

Next, consider the “vector” product, so called because the result is a vector. Given any
vectors A and B, the vector product, written A × B, is defined as:

AB = A B sin n (2.7.1)
×
θ