Engineering Calculus Notes 69

Engineering Calculus Notes 69 - 57 1.5 PLANES 6 Consider...

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Unformatted text preview: 57 1.5. PLANES 6. Consider the line ℓ in the plane defined as the locus of the linear equation in the two rectangular coordinates x and y Ax + By = C. Define the vector − → → → N = A− + B − . ı → (a) Show that ℓ is the set of points P whose position vector − p satisfies →→ −− N · p = C. → (b) Show that if − 0 is the position vector of a specific point on the p → line, then ℓ is the set of points P whose position vector − p satisfies →→→ −−− N · ( p − p 0 ) = 0. → − (c) Show that N is perpendicular to ℓ. 7. Show that if ℓ is a line given by Ax + By = C then the distance from a point Q(x, y ) to ℓ is given by the formula dist(Q, ℓ) = |Ax + By − C | √ . A2 + B 2 (1.20) → − → p (Hint: Let N be the vector given in Exercise 6, and − 0 the position vector of any point P0 on ℓ. Show that −→ − → → →→ dist(Q, ℓ) = proj− P0 Q = proj− (− − − 0 ) , and interpret this in q p N N terms of A, B , C , x and y .) 1.5 Planes Equations of Planes We noted earlier that the locus of a “linear” equation in the three rectangular coordinates x, y and z Ax + By + Cz = D (1.21) ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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