Three Dimensional Geometry–1(Straight Line)

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Session
Three Dimensional Geometry1(Straight Line)
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Session Objectives

• Equation Passing Through a Fixed Point and
  Parallel to a Given Vector

• Equation Passing Through Two Fixed Points

• Co-linearity of Three Points

• Angle Between Two Lines

• Intersection of two lines, Perpendicular
  distance, Image of a Point

• Shortest Distance Between Two Lines

• Class Exercise

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Equation of a Line Passing Through a Fixed Point
and Parallel to a Given Vector
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Cartesian Form
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Example-1
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Example 2
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Solution Cont.
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Passing Through Two Fixed Points
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Cartesian Form
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Example 3
Find the vector and the Cartesian equations for
the line through the points A(3, 4, -7) and
B(5, 1, 6).
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Solution Cont.
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Example 4
Find the coordinates of the points where the line
through A(5, 1, 6) and B(3, 4, 1) crosses the y
z- plane.
Solution The vector equation of the line through
the points A and B is
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Solution Cont.
This point must satisfy (i)
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Co-linearity of Three Points
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Example -5
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Angle Between Two Lines
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Cartesian Form
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Example 6
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Solution Cont.
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Example 7
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Solution Cont.
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Intersection of Two Lines Example - 8
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Solution Cont.
If the lines intersect, then they have a common
point.
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Solution Cont.
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Perpendicular Distance Example 9
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Solution Cont.
Direction ratios of the given line are 2, - 2, -1.
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Solution Cont.
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Image of a Point Example 10
Solution Let Q be the image of the given
point P(2, -1, 5) in the given line and let L
be the foot of the perpendicular from P on the
given line.
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Solution Cont.
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Solution Cont.
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Solution Cont.
Hence, the image of P(2, -1, 5) in the given line
is (0, 5, 1).
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Shortest Distance Between Two Lines
Two straight lines in space, which do not
intersect and are also not parallel, are called
skew lines. (Which do not lie in the same plane)
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Cont.
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Cont.
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Cont.
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Example 11
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Solution Cont.
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Solution Cont.
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Example -12
Solution The lines will intersect if shortest
distance between them 0
Therefore, the given lines intersect.
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