Help : Perpendicular distance of the plane

In summary, to find the perpendicular distance of the plane 5x+2y-z=-22 from the origin O, we first need to find the coordinates of a point P on the plane such that OP is perpendicular to the given plane. One way to do this is by setting x and y equal to zero and solving for z. The perpendicular distance will be the absolute value of this number. Another option is to use Lagrange Multipliers to minimize the distance between the origin and an arbitrary point on the plane, with the constraint that the point must lie on the plane. Another method is to find the normal vector to the plane and use it to write a parametric equation for a line passing through the origin. The intersection of this line with
  • #1
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Homework Statement


Find the perpendicular distance of the plane 5x+2y-z=-22 from the origin O by first finding the co-ordinates of the point P on the plane such that OP is perpendicular to the given plane.


Homework Equations


It only given plane vector,how i going to figure out the perpendicular distance?



The Attempt at a Solution


I really don't know where to start.Can help to elaborate?

Thanks
 
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  • #2
Set x and y equal to zero, solve for z. The perpendicular distance will be the absolute value of this number.
 
  • #3
sandy.bridge said:
Set x and y equal to zero, solve for z. The perpendicular distance will be the absolute value of this number.


You mean (X,Y.Z) = (0.0.Z)?Then minus the plane location?
 
  • #4
One option is to use Lagrange Multipliers to get the coordinates of the point by treating it as a minimization problem (i.e. distance from origin to an arbitrary point) with the constraint that the arbitrary point must lie on the plane. Hint: minimizing the square of the distance also minimizes the distance.
 
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  • #5
You should be able to write a normal to the plane by inspection of the defining equation. Any line that is perpendicular to the plane must be parallel to this normal. So write a parametric equation of a line that passes through the origin that lies along this normal vector. Where does this line intersect the plane?
 

What is the concept of perpendicular distance of a plane?

The perpendicular distance of a plane is the shortest distance between a point and the plane, measured along a line perpendicular to the plane. It is also known as the normal distance or the shortest distance from the point to the plane.

How is the perpendicular distance of a plane calculated?

The perpendicular distance of a plane can be calculated using the formula: d = |ax0 + by0 + cz0 + d| / √(a2 + b2 + c2), where (x0, y0, z0) is the coordinates of the point, and a, b, and c are the coefficients of the plane's equation.

What is the significance of perpendicular distance of a plane in geometry?

The perpendicular distance of a plane is important in geometry as it helps determine the position of a point with respect to a plane. It is also used in various geometric calculations, such as finding the shortest distance between two parallel planes or the distance between a point and a line in space.

Can the perpendicular distance of a plane be negative?

No, the perpendicular distance of a plane is always positive. It represents the shortest distance between a point and a plane, and distance is always a positive quantity.

How is the concept of perpendicular distance of a plane applied in real-life situations?

The concept of perpendicular distance of a plane is applied in various fields such as architecture, engineering, and aviation. It is used to determine the height of buildings, the angle of descent for planes during landing, and the placement of structures on sloped surfaces. It is also used in navigation and surveying to calculate the distance between a point and a reference plane.

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