This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: MATH 20C Lecture 2 - Monday, September 27, 2010 Observations 1. Two vectors pointing in the same direction are scalar multiples of each other. 2. The sum of three head-to-tail vectors in a triangle is 0 . Lines in space Polled the students about what kind of object is described by the equation x +2 y = 5 in the plane. Most got it right (a line). How about the equation x + 2 y + 3 z = 7 in space? It’s a plane (about half of the students answered correctly, the other half thought it was a line). So need some other way to think about lines in space. Think of it as the trajectory of a moving point. Express lines by a parametric equations. In coordinates. For instance, the line through Q = (1 , 2 , 2) and Q 1 = (1 , 3 , 1): moving point Q ( t ) = ( x,y,z ) starts at Q at t = 0, moves at constant speed along line, reaches Q 1 at t = 1. Its “velocity” is ~v =---→ Q Q 1 , so----→ Q Q ( t ) = t---→ Q Q 1 . In our example we get h x +1 ,y- 2 ,z- 2 i = t h 2 , 1 ,- 3 i , i.e. x ( t ) =- 1 + 2 t y ( t ) = 2 + t z ( t ) = 2- 3 t In vector form. The line through a point P in the direction given by some vector ~v is given by ~ r ( t ) =--→ OP + t~v where ~ r ( t ) is the position vector of the point Q ( t ) on the line. Did example with P = (0 , 1 , 1) and v = h , 5 ,- 1 i . (Not sure I remember the exact numbers, I had made it up on the spot in class.) Dot product Definition ~ A · ~ B = a 1 b 1 + a 2 b 2 + a 3 b 3 + ... (a scalar, not a vector) Geometrically ~ A · ~ B = | ~ A || ~ B | cos θ, where θ is the angle between the two vectors....
View Full Document