(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
(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
Transcribed Image Text:### Problem 9
On the last day of class, we described how two curves in \(\mathbb{R}^2\) 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)** Draw a picture of two spheres of different sizes being tangent to each other at exactly one point \( P \) in \(\mathbb{R}^3\). **Do NOT** put one sphere inside the other one.
**(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 for Educational Website
This problem helps illustrate the concept of tangency and how it can be extended from \(\mathbb{R}^2\) (two dimensions) to \(\mathbb{R}^3\) (three dimensions). By examining how normal vectors of surfaces behave, we can understand why and how tangency works in higher dimensions.
**Part (a)** asks students to visualize and draw two spheres that touch at exactly one point. This part emphasizes the spatial understanding of tangent surfaces.
**Part (b)** is a more algebraic approach, asking students to prove the equality of tangent planes using parallel normal vectors. Parallel normal vectors imply that the corresponding tangent planes must coincide. The hint suggests this can be done in a straightforward algebraic step, encouraging the use of fundamental vector operations.
Transcribed Image Text:**Tangent Surfaces and Linearization Functions**
Based on what we have so far, we will say two surfaces \(S_1\) and \(S_2\) 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 \(P\).
(c) If our tangent surfaces are graphs \(S_1 = (\text{graph of } f_1(x,y))\) and \(S_2 = (\text{graph of } f_2(x,y))\), what can you say about the linearization functions \(\mathcal{L}_P f_1(x,y)\) and \(\mathcal{L}_P f_2(x,y)\)? Provide a one-to-two sentence explanation for your claim.
Calculus that deals with differentiation and integration of the vector field in three-dimensional Euclidean space. It deals with quantities that have both magnitude and direction.
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