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midterm_review - Math 20 Midterm Review Carolyn Stein...

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Math 20 Midterm Review Carolyn Stein October 4, 2011 Systems of Equations and the Leontief Model Example: Table 1: A Farming Economy Input for 1 unit Input for 1 unit Input for 1 unit External demand of tomatoes of tomato seeds of labor Tomatoes 0 0.33 0.2 50 Tomato Seeds 0.5 0 0 20 Labor 0.5 0.2 0 0 T = 0(T) + 0.33(S) + 0.2(L) + 50 S = 0.5(T) + 0(S) + 0(L) + 20 L = 0.5(T) + 0.2(S) + 0(L) + 0 What are the units across a row? What is the significance of adding all the numbers in a single row? What are the units across a column? What is the significance of adding all the numbers in a single column? Vectors Geometric Interpretation, Magnitude Vector ~ b = h b 1 , b 2 i is a ray drawn from the origin to the point ( b 1 , b 2 ) Vectors can have up to n dimensions (but we can’t visualize more than three). Using the Pythagorean Theorem, the length or magnitude is: ~ b = k ~ b k = q b 2 1 + b 2 2 + ... + b 2 n 1
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Addition, Multiplication by a Scalar Vector addition adds by components: ~ a + ~ b = h a 1 , a 2 , ..., a n i + h b 1 , b 2 , ..., b n i = h a 1 + b 1 , a 2 + b 2 , ..., a n + b n i Scalar multiplication multiplies each component by a constant: ~ b = h b 1 , b 2 , ..., b n i Dot Product The dot product is the sum of the products of each component. It takes two vectors and returns a constant or scalar - this is why it is also known as the scalar product. ~ a · ~ b = a 1 b 1 + a 2 b 2 Physical Interpretation: The length of ~ a times the length of ~ b that runs in the direction of ~ a (or, equiv- alently, the length of ~ b times the length of ~ a that runs in the direction of ~ b. Picture: Alternative Definition: The physical interpretation allows us to see that ~ a · ~ b = k ~ a kk ~ b k cos where is the angle between the two vectors. Implications: When two vectors are orthogonal ( = 90 o ) the dot product is zero. When two vectors are parallel ( = 0 o ) the dot product is k ~ a kk ~ b k Properties ~ a · ~ b = ~ b · ~ a ~ a · ( ~ b + ~ c ) = ~ a · ~ b + ~ a · ~ c ( ~ a · ~ b ) = k ~ a · ~ b = ~ a · k ~ b ~ a · ~ a = k ~ a k 2 2
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Vector and Parametric Equations of Lines and Planes Line: let P = ( p 1 , p 2 ) be a point on line L, and ~ v = h v 1 , v 2 i be a vector in the direction of L. Then L can be represented by: P + t ~ v (vector) x = p 1 + tv 1 y = p 2 + tv 2 (parametric) Plane: let P = ( p 1 , p 2 , p 3 ) be a point on plane P, and let ~ u = h u 1 , u 2 , u 3 i , ~ v = h v 1 , v 2 , v 3 i be non-parallel vectors on the plane. Then P can be represented by: P + s ~ u + t ~ v (vector) x = p 1 + su 1 + tv 1 y = p 2 + su 2 + tv 2 z = p 3 + su 3 + sv 3 (parametric) Example (2010 Midterm): Let P = (1 , 2 , 1) Q = (4 , 6 , 1) R = (1 , 2 , 3) 1. Find the distance between P and
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