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Problem 710 for which dimensions do both ϕ 1 ϕ 2 ϕ

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PROBLEM 7–10. For which dimensions do both { ϕ 1 , ϕ 2 , . . . , ϕ n } and { ϕ n , ϕ 1 , ϕ 2 , . . . , ϕ n - 1 } have the same orientation? PROBLEM 7–11. For which dimensions do { ϕ 1 , ϕ 2 , . . . , ϕ n } and its “reversal” { ϕ n , . . . , ϕ 2 , ϕ 1 } have the same orientation? Now we consider R 3 and two vectors x , y whose cross product x × y 6 = 0. Writing everything as column vectors, we then have from Section C that det( x y x × y ) = k x × y k 2 > 0 . Thus the frame { x, y, x × y } has the standard orientation. This fact is what settles the choice of direction of the cross product.
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Cross product 11 SUMMARY. The cross product x × y is uniquely determined by the geometric description: (1) x × y is orthogonal to x and y , (2) k x × y k = area of the parallelogram with edges x and y , (3) in case x × y 6 = 0, the frame { x, y, x × y } has the standard orientation. Since the cross product is characterized geometrically, we would expect the action of ele- ments of O (3) to have a predictable outcome. Indeed this is so: THEOREM. Let A O (3). Then A ( x × y ) = (det A ) Ax × Ay. Thus in case A SO(3), then A ( x × y ) = Ax × Ay . PROOF. We use the characterization in terms of column vectors ( x × y ) u = det( x y u ) , all u R 3 . Then for any 3 × 3 real matrix A , ( Ax × Ay ) Au = det( Ax Ay Au ) = det A ( x y u ) = det A det( x y u ) = det A ( x × y ) u. But ( Ax × Ay ) Au = A t ( Ax × Ay ) u. Since we have two vectors with the same inner product with all u R 3 , we thus conclude A t ( Ax × Ay ) = det A ( x × y ) . If A is also invertible, then Ax × Ay = det A ( A t ) - 1 ( x × y ) . Finally, if A O (3), then ( A t ) - 1 = A and det A = ± 1. QED
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12 Chapter 7 The outcome of our discussion is that the cross product enjoys the double-edged properties of being characterized completely geometrically and of being easily computed in coordinates. In Chapter 1 we experienced the same happy situation with dot products on R n . The dot product x y can be completely characterized as k x k k y k cos θ , where θ is the angle between x and y , but also it is easily computed in coordinates. E. Right hand rule This section just gives another way of presenting orientation. Given a Euclidean three- dimensional space, we may think of first choosing an origin and then placing an orthogonal coordinate system x 1 , x 2 , x 3 . We say this is a right-handed coordinate system if when you place your right hand so that your fingers curl from the positive x 1 -axis through 90 degrees to the positive x 2 -axis, your thumb points in the direction of the positive x 3 -axis. Another way of saying this is to place your right hand so that your fingers curl from ˆ ı to ˆ . Then your thumb points in the direction ˆ k if the coordinate system is right-handed. Notice that ˆ ı × ˆ = ˆ k . This then gives the right-hand rule for cross products, supposing we have a right-handed coordinate system. Supposing that x × y 6 = 0, place your right hand so that your fingers curl from x toward y through an acute angle. Then your thumb points in the direction of x × y .
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