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. (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 Suppose we have two surfaces S1 and S2 which intersect at some point P : (a1,b1, c1) (for S1) and (b) (xo, Y0, z0), and further suppose that the normal vectors nı = 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.
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. (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 Suppose we have two surfaces S1 and S2 which intersect at some point P : (a1,b1, c1) (for S1) and (b) (xo, Y0, z0), and further suppose that the normal vectors nı = 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.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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topic: R^3, third dimension, spheres, tangency, drawing models in vector calculus
Transcribed Image Text:### Generalization of Tangency in Multidimensional Space
#### Exercise 9
On the last day of class, we described how two curves in \( R^2 \) being tangent to each other corresponded to their normal vectors being parallel, which helped explain why Lagrange multipliers work. Now, we try to generalize this idea of tangency in various ways.
**(a)** Draw a picture of two spheres of different sizes being tangent to each other at exactly one point \( P \) in \( R^3 \). Do **NOT** put one sphere inside the other one.
*(There is no image provided, so imagine two spheres of different radii in three-dimensional space touching at exactly one point on their surfaces, ensuring neither sphere is inside the other.)*
**(b)** Suppose we have two surfaces \( S_1 \) and \( S_2 \) which intersect at some point \( P = (x_0, y_0, z_0) \), and further suppose that the normal vectors \( \vec{n}_1 = \langle a_1, b_1, c_1 \rangle \) (for \( S_1 \)) and \( \vec{n}_2 = \langle a_2, b_2, c_2 \rangle \) (for \( S_2 \)) at \( P \) are parallel (and non-zero). Prove that the tangent planes \( T_P S_1 \) and \( T_P S_2 \) are the same by transforming the scalar equation for \( T_P S_1 \) into the scalar equation for \( T_P S_2 \).
*Hint: This should be a one-step algebraic transformation based upon using the algebraic definition of parallel vectors.*
**Explanation of Graphs and Diagrams:**
- **Diagram of Spheres (For part (a))**: Imagine two spheres, each with a different radius. These spheres touch at a single point \( P \) in three-dimensional space. At this point, the tangent planes to both spheres coincide, and the normal vectors at this point are collinear.
- **Illustration of Normal Vectors and Tangent Planes (For part (b))**: Consider two surfaces intersecting at point \( P \), with their normal vectors at \( P \) being parallel. The tangent plane to each surface at \( P \) can be
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