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(a) exactly one point P in R³. Do NOT put one sphere inside the other one. Draw a picture of two spheres of different sizes being tangent to each other at

(a) exactly one point P in R³. Do NOT put one sphere inside the other one. Draw a picture of two spheres of different sizes being tangent to each other at

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.4: Plane Curves And Parametric Equations
Problem 33E
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multivariable calculus, vector calculus, vectors

9. On the last day of class, we described how two curves in R? being tangent to each other corresponded to their normal vectors being parallel (and this helped explain why Lagrange multipliers work). Now we try to generalize this idea of tangency in various ways. (а) exactly one point P in R³. Do NOT put oe sphere inside the other one. Draw a picture of two spheres of different sizes being tangent to each other at (b) (xo, Y0, zo), and further suppose that the normal vectors n1 = (a1,b1, c1) (for S1) and n2 = (a2, b2, c2) (for S2) at P are parallel (and non-zero). Prove that the tangent planes TpS1 and TpS2 are the same by transforming the scalar equation for TpSı into the scalar equation for TPS2. Hint: This should be a one-step algebraic transformation based upon using the algebraic definition of parallel vectors. Suppose we have two surfaces S1 and S2 which intersect at some point P
Transcribed Image Text:9. On the last day of class, we described how two curves in R? being tangent to each other corresponded to their normal vectors being parallel (and this helped explain why Lagrange multipliers work). Now we try to generalize this idea of tangency in various ways. (а) exactly one point P in R³. Do NOT put oe sphere inside the other one. Draw a picture of two spheres of different sizes being tangent to each other at (b) (xo, Y0, zo), and further suppose that the normal vectors n1 = (a1,b1, c1) (for S1) and n2 = (a2, b2, c2) (for S2) at P are parallel (and non-zero). Prove that the tangent planes TpS1 and TpS2 are the same by transforming the scalar equation for TpSı into the scalar equation for TPS2. Hint: This should be a one-step algebraic transformation based upon using the algebraic definition of parallel vectors. Suppose we have two surfaces S1 and S2 which intersect at some point P
Based on what we have so far, we will say two surfaces S1 and S2 are tangent to each other at an intersection point P when they have parallel normal vectors at P, which is the same as saying that they have the same tangent plane at P. If our tangent surfaces are graphs S1 = (c) what can you say about the linearization functions Lpf1(x, y) one-to-two sentence explanation for your claim. (graph of f1(x, y)) and S2 = (graph of f2(x, y)), and Lpf2(x, y)? Provide a
Transcribed Image Text:Based on what we have so far, we will say two surfaces S1 and S2 are tangent to each other at an intersection point P when they have parallel normal vectors at P, which is the same as saying that they have the same tangent plane at P. If our tangent surfaces are graphs S1 = (c) what can you say about the linearization functions Lpf1(x, y) one-to-two sentence explanation for your claim. (graph of f1(x, y)) and S2 = (graph of f2(x, y)), and Lpf2(x, y)? Provide a
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