Video narration
How do you turn an arrow on graph paper by ninety degrees and make its length exactly one? Start with a nonzero vector (a,b), pictured as an arrow. A quarter-turn changes it to (-b,a), the perpendicular direction. But rotating does not make it a unit vector; the arrow still has length √(a2+b2). So normalize it-make its length one-by dividing by that length: (-b,a)/√(a2+b2). Check it: the dot product with (a,b) is a times negative b plus b times a, or zero. That proves perpendicularity. Its length is √(b2+a2) over √(a2+b2), which is one. The opposite turn, (b,-a), works too. So the reusable rule for a nonzero two-dimensional vector is: rotate the arrow, then resize it to length one. That answers the question: rotate (a,b) to (-b,a), then divide by √(a2+b2). In three dimensions, an extra choice is needed.