Begin2.DVI

A physical interpretation can be assigned to the triple scalar product A
·(B
×C
)is

that its absolute value represents the volume of the parallelepiped formed by the three

noncoplaner vectors A,
B,
C
when their origins are made to coincide. The absolute

value is needed because sometimes the triple scalar product is negative. This physical

interpretation can be obtained from the following analysis.

In figure 6-9 note the following.

(a) The magnitude |B
×C
|represents the area of the parallelogram P QRS.

(b) The unit vector ˆen=

B
×C


|B
×C
|

is normal to the plane containing the vectors B

and C .

Figure 6-9. Triple scalar product and volume.

(c) The dot product A
·ˆen=A
·

B
×C


|B
×C
|

=h represents the projection of A
on ˆen

and produces the height of the parallelepiped. These results demonstrate that

∣∣

∣A
·(B
×C
)

∣∣

∣=|B
×C
|h=(Area of base)(Height) =Volume.

so that the magnitude of the triple scalar product is the volume of the paral-

lelepiped formed when the origins of the three vectors are made to coincide.

Example 6-7. Show that the triple scalar product satisfies the relations

A
·(B
×C
) = B
·(C
×A
) = C
·(A
×B
)

Note the cyclic rotation of the symbols in the above relations where the first symbol

is moved to the last position and the second and third symbols are each moved to

the left. This is called a cyclic permutation of the symbols.