Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Math 241 Spring 2014 Jayadev S Athreya Spring 2014 Jayadev S Athreya Math 241 Spring 2014 Multiplying 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 a scalar v v1 vn w w1 wn v w v1 w1 v2 w2 vn wn For example if v 2 7 4 w 5 1 3 v w 2 5 7 1 4 3 10 7 12 29 Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Properties of dot product u v w vectors in Rn c R a scalar u v w u v u w cu v c u v u v v u Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 w Angle and Dot Product v v w kvkkwk cos 0 Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Law of Cosines v w w v Law of cosines kw vk2 kwk2 kvk2 2kwkkvk cos Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Magnitudes and Dot Products u u1 un a vector in Rn c R a scalar q Magnitude kuk u12 un2 Scaling kcuk c kuk Dot product u u kuk2 Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Angles and Dot products v w vectors in Rn u w v 0 Law of Cosines kuk2 kwk2 kvk2 2kwkkvk cos Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Angles and Dot products v w vectors in Rn u w v 0 Law of Cosines kuk2 kwk2 kvk2 2kwkkvk cos Dot product kuk2 u u u w v u w u v Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Angles and Dot products v w vectors in Rn u w v 0 Law of Cosines kuk2 kwk2 kvk2 2kwkkvk cos Dot product kuk2 u u u w v u w u v Expanding u w u v w v w w v v w w v w w v v v Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Angles and Dot products v w vectors in Rn u w v 0 Law of Cosines kuk2 kwk2 kvk2 2kwkkvk cos Dot product kuk2 u u u w v u w u v Expanding u w u v w v w w v v w w v w w v v v Dot products kwk2 w w kvk2 v v Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Angles and Dot products v w vectors in Rn u w v 0 Law of Cosines kuk2 kwk2 kvk2 2kwkkvk cos Dot product kuk2 u u u w v u w u v Expanding u w u v w v w w v v w w v w w v v v Dot products kwk2 w w kvk2 v v Magnitudes kuk2 kwk2 kvk2 2w v Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Angles and Dot products v w vectors in Rn u w v 0 Law of Cosines kuk2 kwk2 kvk2 2kwkkvk cos Dot product kuk2 u u u w v u w u v Expanding u w u v w v w w v v w w v w w v v v Dot products kwk2 w w kvk2 v v Magnitudes kuk2 kwk2 kvk2 2w v Equalizing 2w v 2kwkkvk cos Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Angles and Dot products v w vectors in Rn u w v 0 Law of Cosines kuk2 kwk2 kvk2 2kwkkvk cos Dot product kuk2 u u u w v u w u v Expanding u w u v w v w w v v w w v w w v v v Dot products kwk2 w w kvk2 v v Magnitudes kuk2 kwk2 kvk2 2w v Equalizing 2w v 2kwkkvk cos Perpendicular vectors If 2 w v 0 Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Standard Vectors in R2 i 1 0 j 0 1 1 j i 1 u u1 u2 u1 i u2 j Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Standard Vectors in R3 i 1 0 0 j 0 1 0 k 0 0 1 j k i u u1 u2 u3 u1 i u2 j u3 k Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Projection v u v vectors in Rn u 6 0 u proju v proju v is the component of v along u proju v kvk cos u kukkvk cos u u v u kuk kuk kuk kuk2 Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Work Force and Distance Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Describing a Plane A point P0 on the plane a normal vector n perpendicular to the plane For any other point P on the plane the vector P0 P r r0 is perpendicular to n Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Equations for a plane n a b c P0 x0 y0 z0 Then P x y z is on the plane if and only if 0 a b c x x0 y y0 z z0 Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Equations for a plane n a b c P0 x0 y0 z0 Then P x y z is on the plane if and only if 0 a b c x x0 y y0 z z0 That is ax by cz ax0 by0 cz0 Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Equations for a plane n a b c P0 x0 y0 z0 Then P x y z is on the plane if and only if 0 a b c x x0 y y0 z z0 That is ax by cz ax0 by0 cz0 Writing d ax0 by0 cz0 we have the equation of the plane ax by cz d Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product 12 3 Planes in R3 12 5 Example Tangent Plane Jayadev S Athreya Math 241 Spring 2014 Multiplying Vectors Dot product …
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