MAM2000 (Dimension)

Dot Product in Three Dimensions

Geometrically, we know that two vectors are perpendicular if the Pythagorean Theorem holds, i.e. the square of the length of (a,b,c) plus the square of the length of (x,y,z) equals the square of the length of (a,b,c) - (x,y,z) = (a-x,b-y,c-z). This means that a² + b² + c² + x² + y² + z² = (a-x)² + (b-y)² + (c-z)² = a² - 2ax + x² + b² - 2by + y² + c² - 2cz + z². From this it follows that 0 = -2ax - 2by - 2cz, so ax + by + cz = 0. Thus two vectors in R³ are perpendicular if and only if their dot product is zero.

Math Awareness Month 2000 Last modified: 23 Mar 2000 22:45:17 Comments to: thomas_banchoff@brown.edu Hosted by: The Math Forum

Dot Product in Four Dimensions Rotating Space Demo Dot-Product in the Two Dimensions