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3-3 M U LT I P LY I N G V E C T O R S 53

Continuing to expand Eq. 3-26, you can show that

(aybzbyaz)(azbxbzax) (axbybxay). (3-27)

A determinant (Appendix E) or a vector-capable calculator can also be used.
To check whether any xyzcoordinate system is a right-handed coordinate
system, use the right-hand rule for the cross product   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.

iˆ jˆ

iˆ jˆ kˆ

b iˆ jˆ kˆ

:

:a

Checkpoint 5

Vectors and have magnitudes of 3 units and 4 units, respectively. What is the an-

gle between the directions of and if the magnitude of the vector product

is (a) zero and (b) 12 units?

D

:

C

:

D 

:

C

:

C: D:

Figure 3-19Illustration 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 :ab.

:

b

:

:ca:b :a

: b

:

a:

a

b b b

c

a

b

a a

(a)

(b)

c

A