lec6 - distances between points and a line and surfaces in space

8/7/2019 LEC6 - Distances Between Points and a Line and Surfaces in Space

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Distances Between Points, Planes,

and Lines


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This section is concluded with the following discussion oftwo basic types of problems involving distance in space.

1. Finding the distance between a point and a plane

2. Finding the distance between a point and a line

The distance Dbetween a point Qand

a plane is the length of the shortest line

segment connecting Qto the plane,

as shown in Figure 11.52.

Figure 11.52

Distances Between Points, Planes, and Lines


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If Pis anypoint in the plane, you can find this distance byprojecting the vector onto the normal vector n. The

length of this projection is the desired distance.


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Example 5 Finding the Distance Between a Point and a Plane

Find the distance between the point Q(1, 5, 4 ) and theplane given by 3x y+ 2z= 6.

Solution:

You know that is normal to the given plane.

To find a point in the plane, let y= 0 and z= 0 and obtainthe point P(2, 0, 0).

The vector from Pto Qis given by


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Example 5 Solutions

Using the Distance Formula given in Theorem 11.13produces

contd


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From Theorem 11.13, you can determine that the distancebetween the point Q(x0, y0, z0) and the plane given byax+ by+ cz+ d= 0 is

or

where P(x1, y1, z1) is a point in the plane and

d= (ax1 + by1 + cz1).



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Recognize and write equations of cylindricalsurfaces.

Recognize and write equations of quadricsurfaces.

Recognize and write equations of surfaces of

revolution.

Surfaces in Space

Objectives


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Cylindrical Surfaces

This circle is called a generating curve for the cylinder, asindicated in the following definition.


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For the right circular cylinder shown in Figure 11.56, theequation of the generating curve is

x2 + y2 = a2. Equation of generating curve in xy- plane

To find an equation of the cylinder, note that you can generateany one of the rulings by fixing the values of xand yand thenallowing zto take on all real values. In this sense, the value ofzis arbitrary and is, therefore, not included in the equation.

In other words, the equation of this cylinder is simply theequation of its generating curve.

x2 + y2 = a2 Equation of cylinder in space


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Example 1 Solution

b. The graph is a cylinder generated by the sine curve inthe xz-plane.

The rulings are parallel to the y-axis, as shown in

Figure 11.58(b).

Figure 11.58(b)

contd


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Quadric Surfaces


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Quadric Surfaces

The fourth basic type of surface in space is a quadricsurface. Quadric surfaces are the three-dimensionalanalogs of conic sections.


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Quadric Surfaces

The intersection of a surface with a plane is called thetrace of the surface in the plane. To visualize a surface inspace, it is helpful to determine its traces in some well-chosen planes.

The traces of quadric surfaces are conics. These traces,together with the standard form of the equation of eachquadric surface, are shown in the following table.


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Quadric Surfaces


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Quadric Surfacescontd


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Quadric Surfacescontd


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Example 2 Sketching a Quadric Surface

Classify and sketch the surface given by4x2 3y2 + 12z2 + 12 = 0.

Solution:

Begin by writing the equation in standard form.

You can conclude that the surface is a hyperboloid of twosheets with the y-axis as its axis.


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Example 2 Solution

To sketch the graph of this surface, it helps to find thetraces in the coordinate planes.

contd


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Example 2 Solution

The graph is shown in Figure 11.59.

Figure 11.59

contd

lec6 - distances between points and a line and surfaces in space

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