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...Math multivariable calculus spring section the dot product howdowemultiply two vectors we have seen that there is a natural geometric and algebraic way to add but what about multiplying them together it turns out are dierent ways multiply cross this sections focuses on suppose v w let be angle between nd place tails of at same point measure assuming if in direction then they opposite perpendicular give equivalent denitions denition b c cos thus can found by corresponding components adding result or you know lengths along with cosine dimensional as above except cc term vanishes important observation always number scalar example compute answer using note perfectly ne negative value for fact means obtuse because since positive determines sign figure when acute zero exercise i j plane wusing both show agree k orthogonal find nearest degree properties following hold any u these straight forward verify commutative cw cv scalars pull distributive property match wenowshowthat accomplished law ...