From John Taylor's Classical Mechanics:
Show that definition (1.9) of the cross product is equivalent to the elementary deinition that R x S is perpendicular to both R and S, with magnitude rssinθ and direction given by the right hand rule. [Hint: It is a fact (though quite hard to prove) that the definition (1.9) is independent of your choice of axes. Therefore you can choose axes so that R points along the x axis and S lies in the xy plane.
Definition 1.9 refers to the determinant of the 3x3 matrix with the three unit vectors in the top row, vector R components in the second and vector S components in the third.
(Provided in question)
The Attempt at a Solution
I have been able to find the solution when choice of axis is independent by squaring the magnitude of the cross product and using the dot product definition involving the cosine function to eventually arrive at the magnitude equation provided, which I verified with a Khan Academy video that used the same method.
However, I've been trying to do the question in the manner described the question (apologies in advance for the lack of LaTex, still need to learn to use it):
I defined a vector R with only a non-zero x component and a vector S with only x and non-zero y components. When the cross product of R and S is performed, only a z component is left in the new that I called vector T equal to Rx times Sy. This solves the direction/right hand rule part of the question.
To find the magnitude, Pythagoras' theorem is used. For T, the magnitude is equal to Rx times Sy since that's the only component. For R, the magnitude is equal to Rx since that's the only component. This is where I'm running into a problem, how can I make the magnitude of S times sine equal to Sy?
Sorry if that was unclear and thanks in advance for any help.