LECTURE 10
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Planes in 3-Space
In analytic geometry a line in 2-space can be specified by giving its slope and one of its points. Similarly, one can specify a plane in 3-space by giving its inclination and specifying one of its points. A convenient method for describing the inclination of a plane is to specify a nonzero vector, called a normal, that is perpendicular to the plane.
Suppose that we want to find the equation of the plane passing through the point |
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the nonzero vector |
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as a normal. It is evident from Figure 1 that the plane consists |
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precisely of those points |
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for which the vector is orthogonal to n; that is, |
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Since |
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, Equation 1 can be written as |
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(2) |
We call this the point-normal form of the equation of a plane.
Figure 1 Plane with normal vector.
EXAMPLE 1 Finding the Point-Normal Equation of a Plane |
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Find an equation of the plane passing through the point |
and perpendicular to the |
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vector |
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Solution |
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From 2 a point-normal form is
By multiplying out and collecting terms, we can rewrite 2 in the form
where a, b, c, and d are constants, and a, b, and c are not all zero. For example, the equation in Example 1 can be rewritten as
As the next theorem shows, planes in 3-space are represented by equations of the form .
THEOREM 1
If a, b, c, and d are constants and a, b, and c are not all zero, then the graph of the equation
(3)
is a plane having the vector |
as a normal. |
Equation 3 is a linear equation in x, y, and z; it is called the general form of the equation of a plane.
Just as the solutions of a system of linear equations |
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{ |
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correspond to points of intersection of the lines |
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in the |
-plane, so the |
solutions of a system |
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{ |
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(4) |
correspond to the points of intersection of the three planes |
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In Figure 2 we have illustrated the geometric possibilities that occur when (4) has zero, one, or infinitely many solutions.
Figure 2
(a) No solutions (3 parallel planes). (b) No solutions (2 parallel planes). (c) No solutions (3 planes with no common intersection). (d) Infinitely many solutions (3 coincident planes). (e) Infinitely many solutions (3 planes intersecting in a line). (f) One solution (3 planes intersecting at a point). (g) No solutions (2 coincident planes parallel to a third plane). (h) Infinitely many solutions (2 coincident planes intersecting a third plane).
EXAMPLE 2 Equation of a Plane Through Three Points |
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Find the equation of the plane passing through the points |
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Solution
Since the three points lie in the plane, their coordinates must satisfy the general equation of the plane.
Thus
Solving this system gives |
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. Letting |
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desired equation |
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We note that any other choice of t gives a multiple of this equation, so that any value of |
would also give a |
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valid equation of the plane. |
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Alternative Solution |
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Since the points |
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lie in the plane, the vectors |
and |
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are parallel to the plane. Therefore, the equation |
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is normal to the |
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plane, since it is perpendicular to both |
and . From this and the fact that |
lies in the plane, a point- |
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normal form for the equation of |
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the plane is |
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It means, that the equation of plane is |
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Vector Form of Equation of a Plane
Vector notation provides a useful alternative way of writing the point-normal form of the equation of a plane:
Referring to Figure 3, let be the vector from the origin to the point |
, let |
be the vector |
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from the origin to the point |
, and let |
be a vector normal to the plane. |
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Then |
, so Formula 1 can be rewritten as |
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(5)
This is called the vector form of the equation of a plane.
Figure 3
EXAMPLE 3 Vector Equation of a Plane Using 5
The equation
is the vector equation of the plane that passes through the point |
and is perpendicular to the vector |
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Problems Involving Distance
We conclude this section by discussing two basic “distance problems” in 3-space:
Problems
(a)Find the distance between a point and a plane.
(b)Find the distance between two parallel planes.
The two problems are related. If we can find the distance between a point and a plane, then we can find the distance between parallel planes by computing the distance between either one of the planes and an arbitrary point in the other (Figure 4).
Figure 4
The distance between the parallel planes V and W is equal to the distance between and W.
THEOREM 2 |
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Distance Between a Point and a Plane |
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The distance D between a point |
and the plane |
is |
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(6) |
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EXAMPLE 4 Distance Between a Point and a Plane |
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Find the distance D between the point |
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and the plane |
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Solution |
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To apply (6), we first rewrite the equation of the plane in the form |
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Then |
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Given two planes, which are parallel, in this case we can ask for the distance between them. The following example illustrates the latter problem.
EXAMPLE 5 Distance Between Parallel Planes
The planes
are parallel since their normal, |
and |
, are parallel vectors. Find the distance between these |
planes. |
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Solution
To find the distance D between the planes, we may select an arbitrary point in one of the planes and compute its
distance to the other plane. By setting |
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in the equation |
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in this plane. From (6), the distance between |
and the plane |
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is |
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EXAMPLE 6 Angle Between Two Planes
Let’s have two planes
Find the angle between two planes.
Solution |
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To find the angle |
between the planes, we need their normal vectors. |
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Then the angle |
between the planes is the angle between two normal vectors |
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