This preview shows pages 1–14. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 3: Determinants and cross product http://www.math.columbia.edu/~dpt/F10/CalcIII/ September 14, 2010 Announcements I Hand in homework. I Milbank help room is open 10AM–6PM Mon–Thurs, 11AM–3PM Fri. I Extra problems on back of the homework. I Next office hours: Wednesday, 2–3 PM. Lecture 3: Determinants and cross product I Dot product reprise Determinants Cross products Projections: Geometric description The dot product of ~ v and ~ w is ~ v · ~ w = v 1 w 1 + v 2 w 2 + v 3 w 3 = k ~ v kk ~ w k cos ( θ ) . The projection of ~ w onto ~ v , written proj ~ v ( ~ w ) , is the closest point to ~ w on the line containing ~ v . Projections: Geometric description The dot product of ~ v and ~ w is ~ v · ~ w = v 1 w 1 + v 2 w 2 + v 3 w 3 = k ~ v kk ~ w k cos ( θ ) . The projection of ~ w onto ~ v , written proj ~ v ( ~ w ) , is the closest point to ~ w on the line containing ~ v . Projections: Geometric description The dot product of ~ v and ~ w is ~ v · ~ w = v 1 w 1 + v 2 w 2 + v 3 w 3 = k ~ v kk ~ w k cos ( θ ) . The projection of ~ w onto ~ v , written proj ~ v ( ~ w ) , is the closest point to ~ w on the line containing ~ v . Projections and dot product proj ~ v ( ~ w ) is the vector in dir. of ~ v so ~ w proj ~ v ( ~ w ) is perp. to ~ v . proj ~ v ( ~ w ) = a ~ v ( a to be determined) Solve for a : Theorem proj ~ v ( ~ w ) = ~ w · ~ v k ~ v k 2 ~ v . (Note: Scalar times a vector.) Projections and dot product proj ~ v ( ~ w ) is the vector in dir. of ~ v so ~ w proj ~ v ( ~ w ) is perp. to ~ v . proj ~ v ( ~ w ) = a ~ v ( a to be determined) Solve for a : Theorem proj ~ v ( ~ w ) = ~ w · ~ v k ~ v k 2 ~ v . (Note: Scalar times a vector.) Projections and dot product proj ~ v ( ~ w ) is the vector in dir. of ~ v so ~ w proj ~ v ( ~ w ) is perp. to ~ v . proj ~ v ( ~ w ) = a ~ v ( a to be determined) Solve for a : Theorem proj ~ v ( ~ w ) = ~ w · ~ v k ~ v k 2 ~ v . (Note: Scalar times a vector.) Projections and dot product proj ~ v ( ~ w ) is the vector in dir. of ~ v so ~ w proj ~ v ( ~ w ) is perp. to ~ v . proj ~ v ( ~ w ) = a ~ v ( a to be determined) Solve for a : Theorem proj ~ v ( ~ w ) = ~ w · ~ v k ~ v k 2 ~ v . (Note: Scalar times a vector.) Projections and dot product, cont Theorem proj ~ v ( ~ w ) = ~ w · ~ v k ~ v k 2 ~ v . The (signed) length of the projection is comp ~ v ( ~ w ) = ±k proj ~ v ( ~ w ) k = ~ w · ~ v k ~ v k ~ v · ~ w = k ~ v k comp ~ v ( ~ w ) = comp ~ w ( ~ v ) k ~ w k Lecture 3: Determinants and cross product Dot product reprise I Determinants Cross products Matrices and determinants Before getting to the cross product, we’ll look at determinants ....
View
Full
Document
This note was uploaded on 09/26/2011 for the course PHYSICS 106 taught by Professor Arubi during the Summer '11 term at UCLA.
 Summer '11
 Arubi
 Work

Click to edit the document details