Here you can find a proof
this theorem by the method of vector cross products (alternative products)
. The link above also has a java script applet that may help you to visualize
the vector solution.
The Cramer of the rule is Gabriel Cramer (1704-1752), a Swiss mathematician who was a professor of mathematics at Geneva. When Cramer published his rule in 1750 he did not use determinants as they are now shown, and he gave no explanation for how he achieved the result. It seems that Colin Maclaurin probably discovered the same rule as early as 1729, but it was not published until after his death. Although Cramer is primarily remembered for the rule of determinants above he also worked in problems related to physics and general geometry and algebraic curves. It may be that for problems today, Cramer's rule is no longer a practical tool. Henry Thacher, in a review (SIAM News, September 1988) of the Turbo Pascal Numerical Methods Toolbox, writes,"I find it hard to conceive of a situation in which the numerical value of a determinant is needed: Cramer’s rule, because of its inefficiency, is completely impractical, while the magnitude of the determinant is an indication of neither the condition of the matrix nor the accuracy of the solution."
One unusual curve on which he worked is called the Devil's curve. The formula, in cartesian form is given by . The curve is shown in the picture below which is from the University of St Andrews, Scotland webpage on the history of mathematics. A click on the image will take you to their brief biography of Gabriel Cramer.