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Consider a plane II whose unit normal vector pointing outward from the origin is n = ai+ bj+ck and the shortest distance from the origin O(0,0,0) is d, as shown in the figure. Show that all points P(x,y,z) on this plane satisfy the equation ax+ by+cz– d=0. •P d 1=

Consider a plane II whose unit normal vector pointing outward from the origin is n = ai+ bj+ck and the shortest distance from the origin O(0,0,0) is d, as shown in the figure. Show that all points P(x,y,z) on this plane satisfy the equation ax+ by+cz– d=0. •P d 1=

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Consider a plane \( \Pi \) whose unit normal vector pointing outward from the origin is \( \mathbf{n} = ai + bj + ck \) and the shortest distance from the origin \( O(0,0,0) \) is \( d \), as shown in the figure. Show that all points \( P(x,y,z) \) on this plane satisfy the equation \( ax + by + cz - d = 0 \). **Explanation of Diagram:** - The diagram illustrates a three-dimensional coordinate system with a plane \( \Pi \). - The plane \( \Pi \) is positioned such that its normal vector \( \mathbf{n} \) is shown as an arrow pointing outward from the origin \( O(0,0,0) \) to a point on the plane. - The unit normal vector \( \mathbf{n} = ai + bj + ck \) is represented by an arrow perpendicular to the plane, indicating the direction in which it points. - The shortest distance from the origin to the plane is denoted as \( d \), represented by the dashed line connecting the origin \( O \) to a point on the plane along the direction of the normal vector. - The point \( P \) on the plane is a representative point indicating that any point on this plane satisfies the specified equation.
Transcribed Image Text:Consider a plane \( \Pi \) whose unit normal vector pointing outward from the origin is \( \mathbf{n} = ai + bj + ck \) and the shortest distance from the origin \( O(0,0,0) \) is \( d \), as shown in the figure. Show that all points \( P(x,y,z) \) on this plane satisfy the equation \( ax + by + cz - d = 0 \). **Explanation of Diagram:** - The diagram illustrates a three-dimensional coordinate system with a plane \( \Pi \). - The plane \( \Pi \) is positioned such that its normal vector \( \mathbf{n} \) is shown as an arrow pointing outward from the origin \( O(0,0,0) \) to a point on the plane. - The unit normal vector \( \mathbf{n} = ai + bj + ck \) is represented by an arrow perpendicular to the plane, indicating the direction in which it points. - The shortest distance from the origin to the plane is denoted as \( d \), represented by the dashed line connecting the origin \( O \) to a point on the plane along the direction of the normal vector. - The point \( P \) on the plane is a representative point indicating that any point on this plane satisfies the specified equation.
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