Perpendicular Vectors in Rⁿ

Linear Algebra (Matrices) · Axiom Academy

Perpendicular in Any Dimension In 2D you can see a right angle. In you can't — but one number, the dot product, still tells you when two vectors are perpendicular. The right angle, hiding inside one number "Perpendicular" is a picture: two vectors meeting at a clean . That picture works in the plane and in space, but it runs out the moment you reach four dimensions or more — there's nothing left to look at. So we need a test for perpendicular that is pure arithmetic, that agrees with the picture where we can see, and keeps working where we can't. That test is the dot product. Watch what the dot product really measures. Keep fixed and swing down toward a right angle. The shaded bar is the shadow of along — its projection. As the angle approaches that shadow shrinks to a single point, and the dot-product readout falls to exactly 0 . The dot product is the length of that shadow times . It hits 0 exactly when the shadow vanishes — that is, exactly at a right angle. Positive, zero, negative — the sign is the angle One reveal isn't a rule. Watch the same pair sweep through every angle, acute to obtuse. The dot product runs , and it passes through 0 at one place only: the right angle. That's because , and is positive below , zero at , negative above. Below the vectors lean the same way ( + ); above, they lean apart ( - ). Dead perpendicular is the single crossing at 0 . Pythagoras still holds — in four dimensions

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