Vector Concepts: Questions & Answers

[nns91]

I am not sure about these things about vector:

- Is it true to say that a vector n orthogonal to vector n1 and n2 is the cross product of n1 and n2 ?

- Is it true to say that if a line is parallel to a plane, the normal vector of the plane is orthogonal to the parallel vector of the line ?

- Is the acute angle between two planes the angle created by two normal vectors of those planes ?

 

[phreak]

Is it true to say that a vector n orthogonal to vector n1 and n2 is the cross product of n1 and n2 ?

This isn't true, because there are many vectors orthogonal to both n1 and n2 (since the orthogonal complement of {n1,n2} is an entire subspace). In the case of R^3, the subspace is a "line" of vectors, and you can find the vector that is the cross product of n1 and n2 by specifying its norm as |n1|*|n2| and then finding its direction.

 

[csprof2000]

1.) No. There are always an infinite number of vectors which are orthogonal to any two vectors n1 and n2. Only one of these will be the cross product.

2.) Yes. Because a line parallel to a plane would lie in the plane after a suitable translation, and if it lies in the plain, well, it's orthogonal to the plane's normal.

3.)Yes. Assume intersecting but not coincident planes. Their intersection is a line lying in both planes. Now imagine rotating these pictures so that you're looking down this line; the cross section you see will be two intersecting lines, one for each of the two planes. Now, go out a little from their intersection point (in the cross section) and draw two intersecting normal vectors, one originating from one line, one from the other. Then it follows from a simple geometric construction that the acute angle formed by the intersecting normal vectors is, in fact, equal to the angle formed by the plane-lines. I assume this is what you were asking.

 

[nns91]

Thanks.

Then how should I find a vector that orthogonal to both n1 and n2 ?

 

[phreak]

Take their cross product. Don't confuse the statement and its contrapositive.

 

[mathman]

In 3 space, the cross product gives a vector perpendicular to the given pair of vectors. Any scalar multiple of the cross product is also perpendicular to the given pair.

 

[HallsofIvy]

Take their cross product. Don't confuse the statement and its contrapositive.

I think you mean "converse" rather than "contrapositive".

 

[phreak]

Yes, of course. You actually should confuse the statement and its contrapositive, since they're equivalent. Sorry for the confusion.