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.

Projection of any vector x,y,z on the plane along the normal vector a,b,c will be d.

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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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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