3.6. EXERCISES 51

11. Using the notion of the box product yielding either plus or minus the volume of theparallelepiped determined by the given three vectors, show that

(a×b) ·c = a·(b× c)

In other words, the dot and the cross can be switched as long as the order of thevectors remains the same. Hint: There are two ways to do this, by the coordinatedescription of the dot and cross product and by geometric reasoning. It is better ifyou use geometric reasoning.

12. Is a×(b× c) = (a×b)× c? What is the meaning of a×b× c? Explain. Hint: Try(i× j)×j.

13. Discover a vector identity for (u×v)×w and one for u×(v×w).

14. Discover a vector identity for (u×v)× (z×w).

15. Simplify (u×v) · (v×w)× (w× z) .

16. Simplify |u×v|2 +(u ·v)2−|u|2 |v|2 .

17. For u,v,w functions of t, u′ (t) is defined as the limit of the difference quotient as incalculus, (limh→0 w(h))i ≡ limh→0 wi (h) . Show the following

(u×v)′ = u′×v+u×v′, (u ·v)′ = u′ ·v+u ·v′

18. If u is a function of t, and the magnitude |u(t)| is a constant, show from the aboveproblem that the velocity u′ is perpendicular to u.

19. When you have a rotating rigid body with angular velocity vector , then the velocityvector v≡ u′ is given by

v = ×uwhere u is a position vector. The acceleration is the derivative of the velocity. Showthat if is a constant vector, then the acceleration vector a = v′ is given by the for-mula

a = ×( ×u) .Now simplify the expression. It turns out this is centripetal acceleration.

20. Verify directly that the coordinate description of the cross product, a×b has theproperty that it is perpendicular to both a and b. Then show by direct computationthat this coordinate description satisfies

|a×b|2 = |a|2 |b|2− (a ·b)2

= |a|2 |b|2(1− cos2 (θ)

)where θ is the angle included between the two vectors. Explain why |a×b| hasthe correct magnitude. All that is missing is the material about the right hand rule.Verify directly from the coordinate description of the cross product that the rightthing happens with regards to the vectors i, j,k. Next verify that the distributive lawholds for the coordinate description of the cross product. This gives another way toapproach the cross product. First define it in terms of coordinates and then get thegeometric properties from this. However, this approach does not yield the right handrule property very easily.