53
3-3 M ULTIPLYING VECTORS
Continuing to expand Eq. 3-26, you can show that
ϫ ϭ(a y b z Ϫ b y a z ) ϩ (a z b x Ϫ b z a x ) ϩ (a x b y Ϫ b x a y ) .
(3-27)
A determinant (Appendix E) or a vector-capable calculator can also be used.
To check whether any xyz coordinate system is a right-handed coordinate
system, use the right-hand rule for the cross product ϫ ϭ with that system. If
your fingers sweep (positive direction of x) into (positive direction of y) with
the outstretched thumb pointing in the positive direction of z (not the negative
direction), then the system is right-handed.
j
ˆ
i
ˆ
k ˆ
j
ˆ
i
ˆ
k ˆ
j
ˆ
i
ˆ
b
:
a
:
Checkpoint 5
Vectors and have magnitudes of 3 units and 4 units, respectively. What is the angle between the directions of and if the magnitude of the vector product
is (a) zero and (b) 12 units?
D
:
C
: ϫ
D
:
C
:
D
:
C
:
Figure 3-19 Illustration of the right-hand rule for vector products. (a) Sweep vector into vector with the fingers of your right hand.
Your outstretched thumb shows the direction of vector
. (b) Showing that
is the reverse of
.
a
: ϫ b
:
b
: ϫ a
:
c
: ϭ a
: ϫ b
:
b
:
a
:
a
b
b
b
c
a
b
a
a
(a)
(b)
c ؅
A