How to calculate a cross product

[Rocko]

anyone have a clear definition of how to do a cross product?

 

[Galileo]

The cross-product $\vec C$ of two vectors $\vec A$ and $\vec B$ is most conveniently defined by:
$|\vec C|=|\vec A||\vec B|\sin(\theta)$
where $\theta$ is the angle between $\vec A$ and $\vec B$.
This gives the magnitude of $\vec C$. The direction is given by the right-hand rule.

To calculate the cross product when you know the components,
it's usually easiest to form the symbolic 3X3 determinant:
$$\vec A=(A_x,A_y,A_z)$$
$$\vec B=(B_x,B_y,B_z)$$
$$\vec C = \left|<br /> \begin{array}{ccc}<br /> \hat x & \hat y &\hat z \\<br /> A_x & A_y & A_z\\<br /> B_x & B_y & B_z<br /> \end{array}\right|$$
This follows from $\hat x \times \hat y = \hat z$ (and the other possible product combinations of these unit vectors) and the distributivity of the cross product (which is tedious to prove IIRC).

 

[wais]

A cross product is a mathematical operation that takes two vectors as inputs and produces a third vector that is perpendicular to both input vectors. It is also known as a vector product.

To perform a cross product, you need to follow these steps:

1. Identify the two input vectors, let's call them A and B.
2. Write the two vectors in terms of their components in the form of [A1, A2, A3] and [B1, B2, B3].
3. Write the cross product formula: A x B = [A2B3 - A3B2, A3B1 - A1B3, A1B2 - A2B1]
4. Multiply each component of A by the corresponding component of B and subtract the products.
5. The resulting vector is the cross product of A and B.

Keep in mind that the order of the vectors matters in a cross product. Switching the order will result in a vector in the opposite direction.

For example, if A = [2, 3, 4] and B = [5, 6, 7], then the cross product A x B = [-3, 6, -3]. This means that the resulting vector is perpendicular to both A and B, and its direction is determined by the right-hand rule.

In summary, a cross product is a mathematical operation used to find a vector that is perpendicular to two given vectors. It follows a specific formula and the order of the vectors matters.