Kenneth Kuttler

296 CHAPTER 14. VECTOR PRODUCTS

14.4.1 The Box Product

Definition 14.4.8 A parallelepiped determined by the three vectors a,b, and c con-sists of

{ra+ sb+ tc : r,s, t ∈ [0,1]} .That is, if you pick three numbers, r,s, and t each in [0,1] and form ra+ sb+ tc, then thecollection of all such points is what is meant by the parallelepiped determined by thesethree vectors.

The following is a picture of such a thing.

a×b

You notice the area of the base of the parallelepiped, the parallelogram determined bythe vectors a and b has area equal to |a×b| while the altitude of the parallelepiped is|c|cosθ where θ is the angle shown in the picture between c and a×b. Therefore, thevolume of this parallelepiped is the area of the base times the altitude which is just

|a×b| |c|cosθ = a×b ·c.

This expression is known as the box product and is sometimes written as [a,b,c] . Youshould consider what happens if you interchange the b with the c or the a with the c. Youcan see geometrically from drawing pictures that this merely introduces a minus sign. Inany case the box product of three vectors always equals either the volume of the paral-lelepiped determined by the three vectors or else minus this volume.

Example 14.4.9 Find the volume of the parallelepiped determined by the vectors i+2j−5k,i+3j−6k,3i+2j+3k.

According to the above discussion, pick any two of these, take the cross product andthen take the dot product of this with the third of these vectors. The result will be either thedesired volume or minus the desired volume.

(i+2j−5k)× (i+3j−6k) =

∣∣∣∣∣∣i j k1 2 −51 3 −6

∣∣∣∣∣∣= 3i+j+k

Now take the dot product of this vector with the third which yields

(3i+j+k) · (3i+2j+3k) = 9+2+3 = 14.

This shows the volume of this parallelepiped is 14 cubic units.There is a fundamental observation which comes directly from the geometric definitions

of the cross product and the dot product.

296 CHAPTER 14. VECTOR PRODUCTS14.4.1 The Box ProductDefinition 14.4.8 4 parallelepiped determined by the three vectors a,b, and c con-sists of{ra+sb+te:r,s,t € [0,1]}.That is, if you pick three numbers, r,s, and t each in [0,1] and form ra+sb+ tc, then thecollection of all such points is what is meant by the parallelepiped determined by thesethree vectors.The following is a picture of such a thing.AaxbYou notice the area of the base of the parallelepiped, the parallelogram determined bythe vectors a and b has area equal to |a x b| while the altitude of the parallelepiped is|c|cos@ where @ is the angle shown in the picture between c and a x b. Therefore, thevolume of this parallelepiped is the area of the base times the altitude which is just|a x b| |c|cos8 =a x b-e.This expression is known as the box product and is sometimes written as [a,b,c]. Youshould consider what happens if you interchange the b with the c or the a with the c. Youcan see geometrically from drawing pictures that this merely introduces a minus sign. Inany case the box product of three vectors always equals either the volume of the paral-lelepiped determined by the three vectors or else minus this volume.Example 14.4.9 Find the volume of the parallelepiped determined by the vectors 4+ 23 —5k,i+3j —6k,3t +27 +3k.According to the above discussion, pick any two of these, take the cross product andthen take the dot product of this with the third of these vectors. The result will be either thedesired volume or minus the desired volume.ij k(i+2j —5k) x (+37 -6k) =| 1 2 -5 |=3i4+7+k1 3 -6Now take the dot product of this vector with the third which yields(3i+j+k)-(3i+27+3k)=94+243= 14.This shows the volume of this parallelepiped is 14 cubic units.There is a fundamental observation which comes directly from the geometric definitionsof the cross product and the dot product.