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

This preview shows pages 10–13. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern