Vector planes
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The theory of vector planes
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A plane can be specified by: (1)a point lying in the plane (2)a line that is perpendicular to the plane. [Diagram goes here - download the original to see it.] It is not necessary that the line passes through the given point A, and it is sufficient to know the direction of the line - that is, any vector, u, that is parallel to the line. [Diagram goes here - download the original to see it.] This suggests that a plane can be given a vectorial form. This is indeed the case. The general equation of a plane in vector form is [Equation goes here - download the original to see it.] Where is the positio vector of any point lying in the plane and is any vector that is perpendicular to the plane; k is a real number. To prove this, let A be any point in a plane with position vector . Let be a vector perpendicular to [Equation goes here - download the original to see it.]. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] The position vector r of any point A that lies in the plane also lies in the plane and is consequently perpendicular to the vector u that is perpendicular to the plane. Vector planes also have a Cartesian form. That is [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] which gives the Cartesian form of the plane. It is equally easily to exchange from the Cartesian equation to the vector plane. Simply reverse the process. Whilst this embodies the entire theory of vector planes these simple definitions give rise to a large number of problems. (1)To determine whether a line lies in plane, is parallel to a plane, or intersects a plane, and to find the point of intersection of a line and a plane when it exists. If a line lies in a plane, then every point on the line satisfies the vector equation of the plane. [Equation goes here - download the original to see it.] Example [Equation goes here - download the original to see it.] Answer [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Looking at the vector equation of a plane: [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] If a line does not lie in a plane and is not parallel to it either then, in 3 dimensions it must intersect with it. The unique point of intersection can be found by substituting the equation of the line into the equation of the plane and solving for the parameter. It is best to clarify this by an example Example Find the point of intersection of the line [Equation goes here - download the original to see it.] Answer [Equation goes here - download the original to see it.] is the point of intersection of the line with the plane. (2)To find the line of intersection of two non-parallel planes. The following diagram illustrates this problem [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] Equation goes here - download the original to see it.] Diagram goes here - download the original to see it.] Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Example [Equation goes here - download the original to see it.] Answer [Equation goes here - download the original to see it.] then [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] (3)To find the perpendicular distance from a point to a plane: [Diagram goes here - download the original to see it.] The perpendicular length will be a length of a vector that is perpendicular to the plane. [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Example [Equation goes here - download the original to see it. Answer [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] (4) To find the perpendicular distance from a point to a line [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.]Example [Equation goes here - download the original to see it. Solution [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] (5) To find the angle between a line and a plane and the angle between two planes. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] The angle between the line and a plane is defined to be [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] Thus if we are given the vector form of the plane and the line finding the angle between them is almost automatic. Example. [Equation goes here - download the original to see it.] Answer [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it. [Equation goes here - download the original to see it.] Example [Equation goes here - download the original to see it.] Answer [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] (6) To find the shortest distance between two skew lines. In three dimensions two lines are skew when they are not parallel and do not intersect. [Diagram goes here - download the original to see it.][Equation goes here - download the original to see it.][Diagram goes here - download the original to see it.][Equation goes here - download the original to see it.][Equation goes here - download the original to see it.][Equation goes here - download the original to see it.][Equation goes here - download the original to see it.]Example[Equation goes here - download the original to see it.Answer[ Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.][Equation goes here - download the original to see it.][Equation goes here - download the original to see it.]
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Contents of
Vector planes
1 The theory of vector planes
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