2 Notes 1

2 Notes 1 - ~ A and its projection on the xy-plane, then...

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MATH 20C Lecture 1 - Friday, September 24, 2010 Handout: syllabus Vectors A vector (notation: ~ A ) has a direction, and a length ( | ~ A | ). It is represented by a directed line segment, or arrow. In a coordinate system it’s expressed by components. For instance, in space, ~ A = h a 1 ,a 2 ,a 3 i = a 1 ˆ ı + a 2 ˆ + a 3 ˆ k. (Recall in space x -axis points to the lower-left, y -axis to the right, z -axis up.) It tells me in which direction and how far to move. Scalar multiplication Just scale the vector (keep the direction, change the length). Formula for length Drew the vector h 1 , 2 , 3 i and asked for its length. Most students got the right answer ( 14). Explained how why | ~ A | = p a 2 1 + a 2 2 + a 2 3 by reducing to the Pythagorean theorem in the plane. Namely, we first drew a picture showing
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Unformatted text preview: ~ A and its projection on the xy-plane, then derived ~ A from the length of the projection and the Pythagorean theorem (applied twice). Vector addition Drew a picture of the parallelogram with sides ~ A and ~ B and showed how the diagonals are ~ A + ~ B and ~ A-~ B. Addition works componentwise and indeed ~ A = h a 1 ,a 2 ,a 3 i = a 1 ˆ ı + a 2 ˆ + a 3 ˆ k in our earlier example. Application: Used vector additon to find the components of the vector from point P to point Q. Showed that--→ PQ =--→ OQ---→ OP = h q 1-p 1 ,q 2-p 2 ,q 3-p 3 i (where O denotes the origin of the coordinate system). 1...
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This note was uploaded on 04/28/2011 for the course MATH 20C taught by Professor Helton during the Spring '08 term at UCSD.

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