In the following two problems I am trying to get a deeper intuition of, the plane has 3 variables and is 3 dimensional and the hyperplane has four variables and is 3 dimensional as well. Can someone please show me why, practically, in the context of these problems? Question with hyperplane: Find an equation of the hyperplane H in R^4 that passes through P(3, -4, 1, -2) and is normal to u = [2,5,-6,-3]. Question with (regular) plane: Find an equation of the plane H in R^3 that contains P(1,-3,-4) and is parallel to the plane H' determined by the equation 3x - 6y + 5z = 2. Any input would be greatly appreciated!