I understand the solution of a linear system of equation is equivalent to finding the intersection points of n-hyperplanes.
There are 3 elementary row operations - scaling an equation, exchanging equations and subtracting a scalar multiple of an equation from another equation.
The first two I understand geometrically. I am trying to understand the subtraction of two equations geometrically. I can see that the row operation produces a new hyperplane which has one of the coordinate axes as its normal. What I am specifically trying to understand is how the hyperplane is rotated by exactly the right angle?. Anything to do with direction cosines/