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251sol2

# 251sol2 - Section 12.3 Page 1 Section 12.4 Page 1 Section...

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Assignment 2 Multivariate Calculus Math 251 (Fall 2010) A1. Consider the line L in R 2 described parametrically by { r 0 + t v : −∞ < r < ∞} , where r 0 = ( x 0 ,y 0 ) and v = ( a,b ) . i) Write a formula which gives the distance from the origin to a point on L . Your formula should involve only the constants x 0 ,y 0 ,a,b and the variable t . ii) Clearly that there is a single value of t which minimizes the distance from the origin to a point on L . Use the first derivative test (Section 4.7 of text) to find this optimum value of t (in terms of x 0 ,y 0 ,a,b ). iii) Substitute your optimum value of t into the distance formula and simplify this as far as possible, to find the minimum distance from the origin to L . iv) Let v be a vector perpendicular to v and write out an expression for comp v r 0 , the scalar projection of r 0 onto v , in terms of the four constants x 0 ,y 0 ,a,b . v) Explain in your own words why it is not surprising that the answers to parts ii) and iii) are equal, up to sign. Solution i) The distance is f ( t ) = | r 0 + t v | = radicalbig ( x 0 + ta ) 2 + ( y 0 + tb ) 2 . ii) Using the chain rule, we find that f ( t ) = 2 a ( x 0 +
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251sol2 - Section 12.3 Page 1 Section 12.4 Page 1 Section...

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