Math 241, Spring 2014Jayadev S. AthreyaSpring 2014Jayadev S. Athreya Math 241, Spring 2014Vectors (§12.2)··vwVectors have magnitude and directionJayadev S. Athreya Math 241, Spring 2014Vectors in R2··vvIf two vectors have the same magnitude and direction, they areequal.Jayadev S. Athreya Math 241, Spring 2014Adding vectorsHow do we add vectors?··vwJayadev S. Athreya Math 241, Spring 2014Adding vectorsTo form v + w, move to same origin, form paralellogram:·vwv + w·v·wJayadev S. Athreya Math 241, Spring 2014Scaling vectorsGiven c ∈ R, we form the scalar multiple cv, by scalingmagnitude by c. If c < 0, it flips directions.·v·2v·-vJayadev S. Athreya Math 241, Spring 2014Vectors and PointsEach vector v is equal to one with tail at the origin.·vvSo given a point P ∈ R2, we can associate to it the vector−→OP,where O is the origin.We think of vectors and points as different.Jayadev S. Athreya Math 241, Spring 2014Componentsv = (v1, v2), we call v1, v2the components of v.·vv1v2Jayadev S. Athreya Math 241, Spring 2014Arithmetic with componentsv = (v1, v2), w = (w1, w2),v + w = (v1+ w1, v2+ w2)vww1w2v + wv1v2v1+ w1v2+ w2Jayadev S. Athreya Math 241, Spring 2014Arithmetic with componentsv = (v1, v2), cv = (cv1, cv2),v2vv1v22v12v2Jayadev S. Athreya Math 241, Spring 2014Properties of Vectorsu, v, w vectors in R2, c, d ∈ R scalars.Zero vector 0 = (0, 0)v + w = w + vu + (v + w) = (u + v) + wv + 0 = vv + (−v) = 0c(v + w) = cv + cwc(dv) = (cd)v1v = vJayadev S. Athreya Math 241, Spring 2014Magnitudev = (v1, v2), we call v1, v2the components of v.Magnitude of v is given bykvk =qv21+ v22.·vv1v2What theorem are we using?Jayadev S. Athreya Math 241, Spring 2014Vectors in R3, RnCan do exactly the same things in R3, and in fact in Rn. Canadd, scalar multiply, etc. Magnitude of v = (v1, . . . , vn) iskvk =qv21+ v22+ . . . v2n.Jayadev S. Athreya Math 241, Spring 2014Multiplying 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 2014Properties 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 2014Angle and Dot Productvwθv · w = kvkkwk cos θJayadev S. Athreya Math 241, Spring 2014Law of Cosinesvwθw − vLaw of cosines: kw − vk2= kwk2+ kvk2− 2kwkkvk cos θJayadev S. Athreya Math 241, Spring
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