First find the equation of a plane through the point (4, 2, 1) with normal vector = 3i - 2j+ 4k. Next find the equation of the line through this same point (4, 2, 1) and with direction parallel to this normal vector = 37-2j+4k. (this line is called the normal line to the plane) Now graph both this plane and the normal line together. 10 ܝ ܐ ܕ ܝ ܝ ܝ 10 سك
First find the equation of a plane through the point (4, 2, 1) with normal vector = 3i - 2j+ 4k. Next find the equation of the line through this same point (4, 2, 1) and with direction parallel to this normal vector = 37-2j+4k. (this line is called the normal line to the plane) Now graph both this plane and the normal line together. 10 ܝ ܐ ܕ ܝ ܝ ܝ 10 سك
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:### Finding the Equation of a Plane and its Normal Line
#### 1. Equation of a Plane
First, find the **equation of a plane** through the point (4, 2, 1) with normal vector **η** = 3**i** - 2**j** + 4**k**.
#### 2. Equation of the Normal Line
Next, find the **equation of the line** through this **same point** (4, 2, 1) and with direction parallel to this normal vector **η** = 3**i** - 2**j** + 4**k**. (This line is called the **normal line** to the plane.)
#### 3. Graphing the Plane and the Normal Line
Now, graph both this **plane** and the **normal line** together.
#### Description of the Graph
The provided graph is a 3D coordinate system with the x, y, and z axes labeled. The axes intersect at the origin, and each axis is marked with ticks indicating positive and negative directions. In this graph:
- The x-axis ranges from -10 to 10.
- The y-axis ranges from -10 to 10.
- The z-axis ranges from -10 to 10.
*Note: To graph the plane and normal line, use an appropriate 3D graphing tool or software to illustrate their relationships in the coordinate system.*
This exercise involves understanding the geometric representation of planes and lines in three-dimensional space, as well as their equations in vector form.
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