Parallel can mean in the same direction, or it can mean in exactly opposite directions. North and south train lines might run parallel to each other, for exampl

Parallel can mean in the same direction, or it can mean in exactly opposite directions. North and south train lines might run parallel to each other, for example. In plain English, two things are orthogonal if they are unrelated or independent of each other. In linear algebra, this meaning is given a more specific technical meaning of perpendicular; transforming a point p by a vector u that is orthogonal to a basis vector v to get p' means that p and p' have the same multiplier on v (i.e. the contribution of v is independent of transformations orthogonal to v)