50 CHAPTER 3. VECTOR PRODUCTS

From the above description of the cross product and dot product, along with the reduc-tion identity,

(u×v) · (z×w) =

ε i jku jvkε irszrws = (δ jrδ ks−δ jsδ kr)u jvkzrws

= u jvkz jwk−u jvkzkw j

= (u · z)(v ·w)− (u ·w)(v · z)

Example 3.5.6 Simplify u×(u×v) .

The ith component is

ε i jku j (u×v)k = ε i jku jεkrsurvs = εki jεkrsu jurvs

= (δ irδ js−δ jrδ is)u jurvs

= u juiv j−u ju jvi

= (u ·v)ui−|u|2 vi

Henceu×(u×v) = (u ·v)u−|u|2 v

because the ith components of the two sides are equal for any i.

3.6 Exercises1. Show that if a×u = 0 for all unit vectors, u, then a = 0.

2. Find the area of the triangle determined by the three points, (1,2,3) ,(4,2,0) and(−3,2,1) .

3. Find the area of the triangle determined by the three points, (1,0,3) ,(4,1,0) and(−3,1,1) .

4. Find the area of the triangle determined by the three points, (1,2,3) ,(2,3,4) and(3,4,5) . Did something interesting happen here? What does it mean geometrically?

5. Find the area of the parallelogram determined by the vectors, (1,2,3), (3,−2,1) .

6. Find the area of the parallelogram determined by the vectors, (1,0,3), (4,−2,1) .

7. Find the volume of the parallelepiped determined by the vectors, i−7j−5k, i−2j−6k,3i+2j+3k.

8. Suppose a,b, and c are three vectors whose components are all integers. Can youconclude the volume of the parallelepiped determined from these three vectors willalways be an integer?

9. What does it mean geometrically if the box product of three vectors gives zero?

10. Using Problem 9, find an equation of a plane containing the two position vectors,a and b and the point 0. Hint: If (x,y,z) is a point on this plane the volume of theparallelepiped determined by (x,y,z) and the vectors a,b equals 0.