Multiplying Vectors: Dot product (§12.3)Planes in R3 (§12.5)Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Math 241, Spring 2014Jayadev S. AthreyaSpring 2014Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Multiplying vectors: Dot product (§12.3)Given v, w vectors in Rn, can form their dot product, which is ascalar.v = (v1, . . . , vn), w = (w1, . . . , wn),v · w = v1w1+ v2w2+ . . . + vnwnFor example, if v = (2, 7, 4), w = (5, 1, 3),v · w = 2(5) + 7(1) + 4(3) = 10 + 7 + 12 = 29Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Properties of dot productu, v, w vectors in Rn, c ∈ R a scalaru · (v + w) = u · v + u · w(cu) · v = c(u · v)u · v = v · uJayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Angle and Dot Productvwθv · w = kvkkwkcos θ (0 ≤ θ ≤ π).Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Law of Cosinesvwθw − vLaw of cosines: kw − vk2= kwk2+ kvk2− 2kwkkvkcos θJayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Magnitudes and Dot Productsu = (u1, . . . , un) a vector in Rn, c ∈ R a scalarMagnitude kuk =qu21+ . . . + u2nScaling kcuk = |c|kukDot product u · u = kuk2Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Angles and Dot productsv, w vectors in Rn, u = w − v, 0 ≤ θ ≤ πLaw of Cosines kuk2= kwk2+ kvk2− 2kwkkvkcos θDot product kuk2= u · u = u · (w − v) = u · w − u · vExpanding u · w − u · v = (w − v) · w − (w − v) · v =w · w − v · w − w · v + v · vDot products kwk2= w · w, kvk2= v · vMagnitudes kuk2= kwk2+ kvk2− 2w · vEqualizing −2w · v = −2kwkkvkcos θPer pendicular vectors If θ = π/2, w · v = 0.Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Angles and Dot productsv, w vectors in Rn, u = w − v, 0 ≤ θ ≤ πLaw of Cosines kuk2= kwk2+ kvk2− 2kwkkvkcos θDot product kuk2= u · u = u · (w − v) = u · w − u · vExpanding u · w − u · v = (w − v) · w − (w − v) · v =w · w − v · w − w · v + v · vDot products kwk2= w · w, kvk2= v · vMagnitudes kuk2= kwk2+ kvk2− 2w · vEqualizing −2w · v = −2kwkkvkcos θPer pendicular vectors If θ = π/2, w · v = 0.Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Angles and Dot productsv, w vectors in Rn, u = w − v, 0 ≤ θ ≤ πLaw of Cosines kuk2= kwk2+ kvk2− 2kwkkvkcos θDot product kuk2= u · u = u · (w − v) = u · w − u · vExpanding u · w − u · v = (w − v) · w − (w − v) · v =w · w − v · w − w · v + v · vDot products kwk2= w · w, kvk2= v · vMagnitudes kuk2= kwk2+ kvk2− 2w · vEqualizing −2w · v = −2kwkkvkcos θPer pendicular vectors If θ = π/2, w · v = 0.Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Angles and Dot productsv, w vectors in Rn, u = w − v, 0 ≤ θ ≤ πLaw of Cosines kuk2= kwk2+ kvk2− 2kwkkvkcos θDot product kuk2= u · u = u · (w − v) = u · w − u · vExpanding u · w − u · v = (w − v) · w − (w − v) · v =w · w − v · w − w · v + v · vDot products kwk2= w · w, kvk2= v · vMagnitudes kuk2= kwk2+ kvk2− 2w · vEqualizing −2w · v = −2kwkkvkcos θPer pendicular vectors If θ = π/2, w · v = 0.Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Angles and Dot productsv, w vectors in Rn, u = w − v, 0 ≤ θ ≤ πLaw of Cosines kuk2= kwk2+ kvk2− 2kwkkvkcos θDot product kuk2= u · u = u · (w − v) = u · w − u · vExpanding u · w − u · v = (w − v) · w − (w − v) · v =w · w − v · w − w · v + v · vDot products kwk2= w · w, kvk2= v · vMagnitudes kuk2= kwk2+ kvk2− 2w · vEqualizing −2w · v = −2kwkkvkcos θPer pendicular vectors If θ = π/2, w · v = 0.Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Angles and Dot productsv, w vectors in Rn, u = w − v, 0 ≤ θ ≤ πLaw of Cosines kuk2= kwk2+ kvk2− 2kwkkvkcos θDot product kuk2= u · u = u · (w − v) = u · w − u · vExpanding u · w − u · v = (w − v) · w − (w − v) · v =w · w − v · w − w · v + v · vDot products kwk2= w · w, kvk2= v · vMagnitudes kuk2= kwk2+ kvk2− 2w · vEqualizing −2w · v = −2kwkkvkcos θPer pendicular vectors If θ = π/2, w · v = 0.Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Angles and Dot productsv, w vectors in Rn, u = w − v, 0 ≤ θ ≤ πLaw of Cosines kuk2= kwk2+ kvk2− 2kwkkvkcos θDot product kuk2= u · u = u · (w − v) = u · w − u · vExpanding u · w − u · v = (w − v) · w − (w − v) · v =w · w − v · w − w · v + v · vDot products kwk2= w · w, kvk2= v · vMagnitudes kuk2= kwk2+ kvk2− 2w · vEqualizing −2w · v = −2kwkkvkcos θPer pendicular vectors If θ = π/2, w · v = 0.Jayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Standard Vectors in R2i = (1, 0), j = (0, 1)i11ju = (u1, u2) = u1i + u2jJayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Standard Vectors in R3i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)kiju = (u1, u2, u3) = u1i + u2j + u3kJayadev S. Athreya Math 241, Spring 2014Multiplying Vectors: Dot product (§12.3)Planes in R3(§12.5)Projectionu, v vectors in Rn, u 6= 0.·uvproju(v)θproju(v) is the component of v along uproju(v) = kvkcos θukuk=kukkvkcos θkukukuk=u · vkuk2uJayadev S. Athreya Math 241, Spring …
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