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Unformatted text preview: 18.02 LECTURE NOTES, FALL 2010 BJORN POONEN These are an approximation of what was covered in lecture. (Please clear your browser’s cache before reloading this file to make sure you are getting the current version.) 1. September 9 1.1. Vectors. A vector v in R 3 is an ordered triple of real numbers. Example: h 2 , 3 , 5 i . (You can have vectors in R 2 or R 95786 too, if you want.) Geometrically, a vector is an arrow with a length and a direction; its position does not matter. Some vectors have standard names: i := h 1 , , i j := h , 1 , i k := h , , 1 i These are vectors in the directions of the three axes. Also, := h , , i . If P is a point in space, then the position vector P is the vector pointing from (0 , , 0) to P . The length (or magnitude ) of v = h a,b,c i is | v | := √ a 2 + b 2 + c 2 . This formula can be explained by using the Pythagorean theorem twice. A unit vector is a vector of length 1. Addition: h 3 , 1 i + h 1 , 4 i = h 4 , 5 i . Subtraction: h 3 , 1 i - h 1 , 4 i = h 2 ,- 3 i . Geometrically: parallelogram law for +, triangle for- . Important: If A and B are two points, and A and B are their position vectors, then the vector from A to B is B- A , because this is what you have to add to A to get to B . Scalar multiplication:- 10 h 3 , 1 i = h- 30 ,- 10 i . Scalar means number (one uses this word when one wants to emphasize that it is not a vector). Scalar multiplication is a scalar times a vector, and the result is a vector. Geometrically: c v has the same (or opposite) direction as v , but possibly a different length. Two vectors v and w are parallel if one of them is a scalar multiple of the other. How do you write v := h 3 , 4 i as a scalar times a unit vector? Answer: the scalar is the length, which is 5, and the unit vector u is the original vector divided by its length: u = v | v | = 1 | v | v = 1 5 h 3 , 4 i = 3 5 , 4 5 . 1 So, in this example, the desired expression is h 3 , 4 i = 5 length 3 5 , 4 5 unit vector . The same procedure works for any nonzero vector v : v = | v | length v | v | unit vector . Another example : 2 i + 3 j + 5 k = h 2 , 3 , 5 i . Question 1.1. Does the zero vector := h , , i have a direction? I like to say that it has every direction, and hence to say that it is parallel to every other vector, and perpendicular to every other vector. Question 1.2. Suppose that M is the midpoint of segment AB . In terms of the position vectors A and B , what is the position vector M ? To get to M , start at A and go halfway from A to B . The vector from A to B is B- A , so the vector from A to M is 1 2 ( B- A ). Thus M = A + 1 2 ( B- A ) = A + B 2 . Question 1.3. Prove that the midpoints of the sides of a space quadrilateral form a paral- lelogram....
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This note was uploaded on 02/27/2012 for the course 18 02 taught by Professor Preston during the Fall '11 term at MIT.
- Fall '11