142 CHAPTER 8. MATRICES

From the way you multiply a matrix by a vector, it follows that Pu (ei) gives the ith columnof the desired matrix. Therefore, it is only necessary to find

Pu (ei)≡( ei·uu ·u

)u

For the given vector in the example, this implies the columns of the desired matrix are

114

 123

 ,214

 123

 ,3

14

 123

 .

Hence the matrix is

114

 1 2 32 4 63 6 9

 .

8.5.3 Rotations About A Particular VectorThe problem is to find the matrix of the linear transformation which rotates all vectorsabout a given unit vector u which is possibly not one of the coordinate vectors i,j, or k.Suppose for |c| ̸= 1

u= (a,b,c) ,√

a2 +b2 + c2 = 1.

First I will produce a matrix which maps u to k such that the right handed rotationabout k corresponds to the right handed rotation about u. Then I will rotate about k andfinally, I will multiply by the inverse of the first matrix to get the desired result.

To begin, find vectors w,v such that w×v = u. Let

w =

(− b√

a2 +b2,

a√a2 +b2

,0).

uw

This vector is clearly perpendicular to u. Then v = (a,b,c)×w ≡ u×w. Thus fromthe geometric description of the cross product, w×v = u. Computing the cross productgives

v = (a,b,c)×(− b√

a2 +b2,

a√a2 +b2

,0)

=

(−c

a√(a2 +b2)

,−cb√

(a2 +b2),

a2√(a2 +b2)

+b2√

(a2 +b2)

)Now I want to have Tw= i,Tv = j,Tu= k. What does this? It is the inverse of the

matrix which takes i to w, j to v, and k to u. This matrix is