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Unformatted text preview: Dot (Scalar, Inner) Product
Given two vectors u and v , the dot (scalar, inner) product between them is deﬁned as: 1. Analytic Deﬁnition: Let u = (x1 , y1 , z1 ) and v = (x2 , y2 , z2 ). Then u · v = x1 x2 + y1 y2 + z1 z2 . 2. Geometric Deﬁnition u·v = u v cos θ where θ is the angle between u and v . Note: given two vectors, the dot product results in a scalar (a real number). Properties of Dot Product
u·v u · (u + w ) αu · v O ·u = = = = v ·u u·v +u·w u · αv 0 Note 1: u · u = u 2 . Note 2: the angle θ between two non-zero vectors can be expressed in terms of the dot product as:
cos θ = u·v , uv or θ = cos−1 u·v , uv Note 3: two (non-zero) vectors are perpendicular (or orthogonal, θ = π , cos θ = 0) if and only if 2 their dot product equals zero. Orthogonal Projection
Given u = O and v , the orthogonal projection of v onto u is given by: Proju v = v ·u u
2 u= v· u u u = (v · u ) u ˆˆ u The scalar projection (or component) of v onto u is given by: Compu v = Note: v − Proju v ⊥ u , or equivalently (v − Proju v ) · u = 0 v· u u = (v · u ) ˆ ...
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