Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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![### Question 5
If \( \mathbf{r} = \langle x, y, z \rangle \) and \( \mathbf{r_0} = \langle x_0, y_0, z_0 \rangle \), describe precisely the set of all points \( (x, y, z) \) such that
\[ |\mathbf{r} - \mathbf{r_0}| = 1 \]
### Answer Explanation
This question asks you to describe the set of all points \((x, y, z)\) that are at a distance of 1 unit from a given point \((x_0, y_0, z_0)\). This set of points forms a sphere with radius 1 centered at the point \((x_0, y_0, z_0)\).
In mathematical terms, the distance between any point \((x, y, z)\) and a fixed point \((x_0, y_0, z_0)\) in three-dimensional space is given by:
\[ ((x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2)^{1/2} \]
For the distance to be equal to 1,
\[ \sqrt{(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2} = 1 \]
Squaring both sides, we get:
\[ (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = 1 \]
Thus, the set of all points \( (x, y, z) \) that satisfy this equation forms a sphere with radius 1 centered at \((x_0, y_0, z_0)\).
This question illustrates the concept of the equation of a sphere and is fundamental in understanding the geometry of spheres in three-dimensional space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F333de6cc-3da3-4bcc-9995-6ee1769198c2%2F4e1e2184-ac55-41e2-8594-0c73ae87dc98%2F2cq6qes.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 5
If \( \mathbf{r} = \langle x, y, z \rangle \) and \( \mathbf{r_0} = \langle x_0, y_0, z_0 \rangle \), describe precisely the set of all points \( (x, y, z) \) such that
\[ |\mathbf{r} - \mathbf{r_0}| = 1 \]
### Answer Explanation
This question asks you to describe the set of all points \((x, y, z)\) that are at a distance of 1 unit from a given point \((x_0, y_0, z_0)\). This set of points forms a sphere with radius 1 centered at the point \((x_0, y_0, z_0)\).
In mathematical terms, the distance between any point \((x, y, z)\) and a fixed point \((x_0, y_0, z_0)\) in three-dimensional space is given by:
\[ ((x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2)^{1/2} \]
For the distance to be equal to 1,
\[ \sqrt{(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2} = 1 \]
Squaring both sides, we get:
\[ (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = 1 \]
Thus, the set of all points \( (x, y, z) \) that satisfy this equation forms a sphere with radius 1 centered at \((x_0, y_0, z_0)\).
This question illustrates the concept of the equation of a sphere and is fundamental in understanding the geometry of spheres in three-dimensional space.
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