102 CHAPTER 6. VECTOR PRODUCTS
22. 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.
23. Discover a vector identity for u×(v×w) .
24. Discover a vector identity for (u×v) · (z×w) .
25. Discover a vector identity for (u×v)× (z×w) in terms of box products.
26. Simplify (u×v) · (v×w)× (w×z) .
27. Simplify |u×v|2 +(u ·v)2−|u|2 |v|2 .
28. Prove that ε i jkε i jr = 2δ kr.
29. If A is a 3×3 matrix such that A =(
u v w)
where these are the columns ofthe matrix A. Show that det(A) = ε i jkuiv jwk.
30. If A is a 3×3 matrix, show εrps det(A) = ε i jkAriAp jAsk.
31. Suppose A is a 3×3 matrix and det(A) ̸= 0. Show using 30 and 28 that
(A−1)
ks =1
2det(A)εrpsε i jkAp jAri.
32. When you have a rotating rigid body with angular velocity vector Ω then the velocity,u′ is given by u′ = Ω×u. It turns out that all the usual calculus rules such as theproduct rule hold. Also, u′′ is the acceleration. Show using the product rule that forΩ a constant vector
u′′ =Ω×(Ω×u) .
It turns out this is the centripetal acceleration. Note how it involves cross products.
33. Find the planes which go through the following collections of three points. In casethe plane is not well defined, explain why.
(a) (1,2,0) ,(2,−1,1) ,(3,1,1)
(b) (3,1,0) ,(2,1,1) ,(−3,1,−1)
(c) (2,1,1) ,(−2,3,1) ,(0,4,2)
(d) (1,0,1) ,(2,0,1) ,(0,1,1)