Talk:Cross product

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I removed from the basic properties of a cross product, as it only applies for three vectors which are orthogonal to each other. — Preceding unsigned comment added by 72.83.127.158 (talkcontribs)

The Cross Product[change source]

Besides the usual addition of vectors and multiplication of vectors by scalars, there are also two types of multiplication of vectors by other vectors. One type, the dot product, is a scalar product; the result of the dot product of two vectors is a scalar. The other type, called the cross product, is a vector product since it yields another vector rather than a scalar. As with the dot product, the cross product of two vectors contains valuable information about the two vectors themselves.

The cross product of two vectors a=<a_1,a_2,a_3> and b=<b_1,b_2,b_3> is given by

Although this may seem like a strange definition, its useful properties will soon become evident. There is an easy way to remember the formula for the cross product by using the properties of determinants. Recall that the determinant of a 2x2 matrix is

and the determinant of a 3x3 matrix is

Notice that we may now write the formula for the cross product as

Example

The cross product of the vectors a=<3,-2,-2> and b=<-1,0,5> is

Properties of the Cross Product:

   * The length of the cross product of two vectors is
   * The length of the cross product of two vectors is equal to the area of the parallelogram determined by the two vectors (see figure below).
   * Anticommutativity:
   * Multiplication by scalars:
   * Distributivity:
   * The scalar triple product of the vectors a, b, and c:
   * The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product.
   * The vector triple product of the vectors a, b, and c: 

Note that the result for the length of the cross product leads directly to the fact that two vectors are parallel if and only if their cross product is the zero vector. This is true since two vectors are parallel if and only if the angle between them is 0 degrees (or 180 degrees).

Example

To find the area of the triangle with vertices (1,1,3), (4,-1,1), and (0,1,8), one could find the length of one of the altitudes of the triangle and proceed to find A=1/2(altitude)(base). However, this may not be an easy task in three dimensions. So consider finding the one-half the area of the parallelogram determined by the vector a from (1,1,3) to (4,-1,1) and the vector b from (1,1,3) to (0,1,8). Then a=<2-1,2-(-1),2-3>=<3,-2,-2> and b=<-1,0,5>. Using the result from the previous example and property number two above, we have the length of the cross product (and therefore the parallelogram determined by a and b) as

So, the area of the traingle is one-half this quantity, or 8.26.

The cross product occurs in many formulas in physics. Some examples include the curl of a vector field (see also Stoke's Theorem), torque, and many integrals over surfaces. — Preceding unsigned comment added by 75.190.237.110 (talkcontribs)

I would like to include the non-vector definition, and place it first.[change source]

OpenStax College Physics is a college course that does not require calculus. They do not mention "cross product", but instead define it using the right hand rule, with the magnitude containing the sine theta term. It seems to me that this should be the first "simple" definition a reader should see of the cross product.--Guy vandegrift (talk) 01:13, 27 February 2018 (UTC)[reply]

Page isn't simple and fails to explain it[change source]

This article definitely doesn't fall under "Simple" as it stands now. In fact, it's really *really* bad. a) It's using a word like "Perpendicular" - yes it may be the right word but it's not a simple word. b) The definition isn't clear. Despite having a masters degree in a technical subject I still don't understand what a Cross Product is. Given that the entire purpose of the article being to explain that, self-evidently the article is failing at its purpose. c) A wall full of equations isn't simple either; if I wanted that I'd be reading the only-suitable-for-a-maths/physics-grads article on en.wikipedia.org d) The maths notations are not Simple. I have no idea what an arrow above a number means, in fact I don't think I've ever even seen it before - it's not used on the en.wikipedia version of the article. Given this is Simple English, it's not unreasonable to also expect Simple Maths. I would suggest that means nothing more than BODMAS - https://en.wikipedia.org/wiki/BODMAS.