Prove the validity of cross products by example/words. (Why does it work as a strategy?)
Is this right?: One way to find out whether ratios are equal, is to find a common denominator. Ratios are equal if their numerator and denominator are equal. It is a lot easier and faster to use cross products than simplifying.
The concept of cross products is not directly related to finding equal ratios by using a common denominator. Cross products are actually a mathematical operation used to find the vector perpendicular to two given vectors in three dimensions.
To understand how cross products work, let's consider an example. Suppose we have two vectors, A = (2, 1, 3) and B = (4, -2, 1). To find their cross product, denoted as A x B, we follow these steps:
1. Write the vectors in matrix form:
A = |2| B = |4|
|1| |-2|
|3| |1|
2. Calculate the determinant of the 2x2 matrix formed by the non-zero terms of the vectors. In this case, it is:
det(A) = (2 * 1) - (3 * -2) = 2 + 6 = 8
3. Create the new vector C whose components are the determinants of the 2x2 matrices formed by the non-zero terms of the original vectors, without their respective rows and columns. In this case, it is:
C = |-2| = (-2) C = |1| = 1
|3| |4|
4. Assign the correct signs to the components of C based on the alternating pattern: positive, negative, positive, etc. In this case, it is:
C = (-2, -1, 1)
Thus, the cross product of A and B is C = (-2, -1, 1).
Now, regarding the statement you provided about finding equal ratios, it is not directly related to the concept of cross products. Comparing ratios to determine equality is a separate concept and can be done by finding a common denominator or using cross multiplication. Cross multiplication, not cross products, is a method used to determine the equality of ratios, as you mentioned in your statement.
In summary, cross products are fundamental in vector algebra and are used to find a new vector perpendicular to two given vectors in three dimensions. They are not directly related to finding equal ratios.