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MIT 18 02 - Topic 17: Vectors, dot product

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18.02A Topic 17: Vectors, dot product.Read: TB: 17.3, 18.1, 18.2VectorsTwo views: First the geometric and then the analytic.Geometric viewVector = length and direction: (Discuss scaling, scalars)????(same vector)Length: denoted |A|, also called magnitude or normAddition: (head to tail) Subtraction: either tail to tail or A+(−B)??A//B77oooooooooooooA + B??A//−B//BTT))))))))A − BTT))))))))Analytic or algebraic viewPlace the tail of A at the origin ⇒ the coordinates of the head determine A:A = ha1, a2i = a1i + a2j.(a1, a2)??A//a1iOOa2j??−→P Q•PQYou’ve seen the vectors i and j in physics. Theyhave coordinates i = h1, 0i, j = h0, 1i//iOOjNotation1, (a1, a2) indicate as point in the plane.2. ha1, a2i indicates the vector from the origin to the point (a1, a2). Of course, thisvector can be translated anywhere and ha1, a2i = a1i + a2j.3.−→P =−−→OP is the vector from the origin to P .4. In print we will often drop the arrow and just use the bold face to indicate a vector,i.e. P ≡−→P.Discuss scaling, scalarsLength: |A| =pa21+ a22Addition: (a1i + a2j) + (b1i + b2j) = (a1+ b1)i + (a2+ b2)j⇔ ha1, a2i + hb1, b2i = ha1+ b1, a2+ b2iDiscuss−−→PQ =−→Q −−→P –geom. and anal(continued)118.02A topic 17 2Dot product (scalar product)Geometric definition:A · B = |A||B|cos θ??A//Bθ(algebraic view)A · B = a1b1+ a2b2(Hard to get geometrically)proof: Law of cosines: (won’t do in class)//B??ATT)))))))))A − Bθ|A − B|2= |A|2+ |B|2− 2|A||B|cos θ⇒ (a21+ a22) + (b21+ b22) − ((a1− b1)2+ (a2− b2)2)= 2|A||B|cos θ⇒ a1b1+ a2b2= |A||B|cos θ. QEDAlgebraic law: A ·(B + C) = A · B + A · C.Follows from the algebraic view of dot product.Unit vectorsSpecial vectors: i and j. Note i · i = 1 = j · j and i · j = 0.Unit vector: u: |u| = 1. Often indicate bybu.A · u = |A|cos θA · A = |A|2A ⊥ B ⇔ A · B = 0//bu;;xxxxxxxxxxxxxxA|A|cos θ = component of A indirection ofbuθComponents or projection:The component of A in the direction ofbu is A ·buFor a non-unit vector: the component of A in the direction of B is the component ofA in the direction ofbu =B|B|.Trig identity cos(β − α) = cos α cos β − sin α sin βUnit vectors: u = cos α i + sin α j, v = cos β i + sin β jAngle between them is θ = β − αGeometric: u · v = |u||v|cos θ = cos θ = cos(β − α)Analytic: u · v = u1v1+ u2v2= cos α cos β + sin α sin β.77oooooooooGGαβ − α(cos α, sin α)(cos β, sin β)Example: P = (−5, 0), Q = (1, 3) ⇒−−→PQ = 6i + 3j = h6, 3i.Example: Show−−→PQ +−−→QR +−−→QP = 077ooooooooo__??????PQR(continued)18.02A topic 17 3Example: Find 2 unit vectors parallel to v = 3i − 4j.|v| = 5: u1=34i −45j, u2= −u1.Example: Show the sum of the medians of a triangle = 0.Median of AB = P =12(A + B) ⇒ CP =12(B + A) − C.Likewise:−−→BQ =12(A + C) − B,−−→AR =12(B + C) − A.⇒ Sum of medians =−−→CP +−−→BQ +−−→AR = 0.ooooooooo•??????••ABCPRQExample: Let A = (1, 2), B = (2, 3) and C = (2, −1). Find the cosine of ∠BAC.Let θ b e the angle ⇒ cos θ =−−→AB ·−−→AC|AB||AC|.−−→AB = h1, 1i,−−→AC = h1, −3i⇒ cos θ =1 − 3√2√10= −2√20=1√5.<<yyyyyyyyyyy //ABCθExample: Velocities are vectorsA river flows at 3mph and a rower rows at 6mph. What heading shouldhe use to get straight across a river?Need sin θ =36⇒ θπ/6Answer: Head at angle of π/6 radians upstream from straigh across.__????????????//OOθSame question with river=2 mph, row=2√2 mph:⇒ sin θ =22√2⇒ θ = π/4.Same question with river=6 mph, row=3 mph:⇒ sin θ =63⇒ No such θThree dimensionsExactly the same except third coordinate: a1i + a2j + a3k = (a1, a2, a3)Example: Show A = (4, 3, 6), B = (−2, 0, 8), C = (1, 5, 0) are the ve rtices of aright triangle.Two legs of the triangle are−−→AC = h−3, 2, −6iand−−→AB = h−6, −3, 2i⇒−−→AC ·−−→AB = 18 − 6 − 12 = 0 ⇒


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MIT 18 02 - Topic 17: Vectors, dot product

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