VIPSolutionsgive answer in 2 step with explanation at the end of each step and final answer at the end:Ex 16.6.1 Describe or sketch the surface with the given vector function. a.r(u,v) = (u+v,3 —v,1+ 4u +50) b. r(u,v) = (2sinu,3 cos u,v) c. x(s,t) = (s5,t,8% — 5%) d. r(s,t) = (ssin2t, s% s cos 2t) Ex 16.6.2 Find a vector function r(u,v) for the surface. a. The plane that passes through the point (1,2, —3) and is parallel to the vectors (1,1,—1) and (1,—1,1). b. The lower half of the ellipsoid 222 + 4y? + 22 = 1. c. The part of the sphere of radius 4 centered at the origin that lies between the planes z=-2andz=2.
Question:
give answer in 2 step with explanation at the end of each step and final answer at the end:
Ex 16.6.1 Describe or sketch the surface with the given vector function.
a.r(u,v) = (u+v,3 —v,1+ 4u +50)
b. r(u,v) = (2sinu,3 cos u,v)
c. x(s,t) = (s5,t,8% — 5%)
d. r(s,t) = (ssin2t, s% s cos 2t)
Ex 16.6.2 Find a vector function r(u,v) for the surface.
a. The plane that passes through the point (1,2, —3) and is parallel to the vectors
(1,1,—1) and (1,—1,1).
b. The lower half of the ellipsoid 222 + 4y? + 22 = 1.
c. The part of the sphere of radius 4 centered at the origin that lies between the planes
z=-2andz=2.
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Asked by: Rahul
Created at: 2025-11-21 03:26:14
Asked by: Rahul
Created at: 2025-11-21 03:26:14
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Let's solve **Ex 16.6.2 (a)** step-by-step as requested:
---
## Ex 16.6.2 (a)
**Find a vector function** $\mathbf{r}(u,v)$ **for the plane that passes through the point** $(1,2,-3)$ **and is parallel to the vectors** $\langle 1,1,-1 \rangle$ **and** $\langle 1,-1,1 \rangle$.
---
### **Step 1: Write the general parametric equation of a plane**
A plane passing through point $\mathbf{r}_ = (x_, y_, z_)$ and parallel to vectors $\mathbf{a}$ and $\mathbf{b}$ is:
\[
\mathbf{r}(u, v) = \mathbf{r}_ + u\mathbf{a} + v\mathbf{b}
\]
**Explanation:**
This equation gives all points on the plane as you vary $u$ and $v$.
---
### **Step 2: Substitute the given values**
Given:
- $\mathbf{r}_ = (1, 2, -3)$
- $\mathbf{a} = (1, 1, -1)$
- $\mathbf{b} = (1, -1, 1)$
Plug into the formula:
\[
\mathbf{r}(u, v) = (1, 2, -3) + u(1, 1, -1) + v(1, -1, 1)
\]
\[
= (1 + u + v,\ 2 + u - v,\ -3 - u + v)
\]
**Explanation:**
We simply add the scaled direction vectors to the point to get the parametric form.
---
## **Final Answer**
\[
\boxed{
\mathbf{r}(u, v) = (1 + u + v,\ 2 + u - v,\ -3 - u + v)
}
\]
This is the required vector function for the plane.
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