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Intersection of Lines and Planes Help (page 2)

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

Adjacent Angles Between An Intersecting Line And Plane

Suppose a line and a plane intersect, and their angles of intersection are u and v as defined above. Then if u and v are specified in degrees, u + v = 180°. If u and v are specified in radians, then u + v = π .

Dropping A Normal To A Plane

Suppose that R is a point near, but not in, a plane X . Then there is exactly one line N through point R , intersecting plane X at some point S , such that line N is normal to plane X , as shown in Fig. 7-10. Any lines in plane X that pass through point S , such as L and M shown in the figure, must necessarily be perpendicular to line N .

 

An Expanded Set of Rules Angles and Distances Dropping A Normal To A Plane

Fig. 7-10 . Line N through point R is normal to plane X at point S . The distance between R and X is equal to the length of line segment RS .

Distance Between A Point And Plane

Suppose that R is a point near, but not in, a plane X . Let N be the line through R that is normal to plane X . Suppose line N intersects plane X at point S . Then the distance between point R and plane X is equal to the length of line segment RS (Fig. 7-10).

Plane Perpendicular To Line

Imagine a line N in space. Imagine a specific point S on line N . There is exactly one plane X containing point S , such that line N is normal to plane X at point S (Fig. 7-10). Any lines in plane X that pass through point S , such as L and M shown in the figure, must necessarily be perpendicular to line N .

Line Parallel To Plane

A line L is parallel to a plane X if and only if the following two conditions hold true:

  • An Expanded Set of Rules Angles and Distances Line Parallel To Plane Line L does not lie in plane X
  • An Expanded Set of Rules Angles and Distances Line Parallel To Plane Line L does not intersect plane X
  • nder these conditions, there is exactly one line M in plane X , such that lines L and M are parallel. Any line N in plane X , other than line M , is a skew line relative to L (Fig. 7-11).

 

An Expanded Set of Rules Angles and Distances Line Parallel To Plane

Fig. 7-11 . A line L parallel to a plane X . There is exactly one line M in plane X such that M is parallel to L; all other lines N in plane X are skew lines relative to L .

Distance Between A Parallel Line And Plane

Suppose line L is parallel to plane X . Let R be some (any) point on line L . Then the distance between line L and plane X is equal to the distance between point R and plane X , which has already been defined.

Addition And Subtraction Of Angles Between Intersecting Planes

Angles between intersecting planes add and subtract just like angles between intersecting lines (or line segments). Here is how we can prove it, based on knowledge we already have.

Suppose three planes X, Y , and Z intersect in a single, common line L . Let S be a point on line L . Let P, Q , and R be points on planes X, Y , and Z respectively, such that line segments SP, SQ , and SR are all perpendicular to line L . Let ∠XY be the angle between planes X and Y, ∠YZ be the angle between planes Y and Z , and ∠XZ be the angle between planes X and Z . This is diagrammed in Fig. 7-12. From the preceding definition of the angle between two planes, we know that:

∠XY = ∠PSQ

∠YZ = ∠QSR

∠XZ = ∠PSR

 

An Expanded Set of Rules Angles and Distances Addition And Subtraction Of Angles Between Intersecting Planes

Fig. 7-12 . Addition and subtraction of angles between planes.

Line segments SP, SQ , and SR all lie in a single plane because they all intersect at point S and they are all perpendicular to line L . Therefore, we know from the rules for addition of angles in a plane, that the following hold true for the measures of the angles between the line segments:

∠PSQ + ∠QSR = ∠PSR

∠PSR∠QSR = ∠PSQ

∠PSR∠PSQ = ∠QSR

Substituting the angles between the planes for the angles between the line segments, we see that the following hold true for the measures of the angles between the planes:

∠XY + ∠YZ = ∠XZ

∠XZ∠YZ = ∠XY

∠XZ∠XY = ∠YZ

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