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3 Notes 2 - MATH 20C Lecture 2 Monday Observations 1 Two...

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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 0 = (1 , 2 , 2) and Q 1 = (1 , 3 , 1): moving point Q ( t ) = ( x, y, z ) starts at Q 0 at t = 0, moves at constant speed along line, reaches Q 1 at t = 1. Its “velocity” is ~v = ---→ Q 0 Q 1 , so ----→ Q 0 Q ( t ) = t ---→ Q 0 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 0 , 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. Explained the result as follows. First, ~ A · ~ A = | ~ A | 2 cos 0 = | ~ A | 2 is consistent with the definition.
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