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Straight line passing through a point and parallel to given vector

Vector form A line passing through point 87127.png and parallel to 87133.png, where λ is parameter (scalar) where 87139.png = position vector of any point on the line.
 
Cartesian form Equation of line passing through the point (x1, y1, z1) and having direction ratios a, b, c is 87149.png (parameter).
 
Any point on the line is (x1 + λa, y1 + λb, z1 + λc)

Line passing through two given points

Vector form The vector equation of a line passing through two points whose position vectors 87155.png is
87161.png
 
Cartesian form Equation of straight line passing through (x1, y1, z1) and (x2, y2, z2) is
87167.png

 

Notes:
  • Two straight lines in a space which are neither parallel nor intersecting are called skew-lines.
     
    Thus, the skew-lines are those lines which do not lie in the same plane.
     
    For lines 90737.png and 90731.png or lines
     
    90724.png
     
    and 90718.png
     
    where 90712.png90706.png,
     
    90700.png90694.png
  • Angle between the lines is cos–190688.png
  • Lines are parallel if 90681.png, where t is scalar.
  • Lines are perpendicular if 90675.png = 0.
  • Lines are coplanar if 90669.png = 0
  • Shortest distance between the lines is 90663.png.

Foot of perpendicular from point on the line

Let “L” be the foot of perpendicular drawn from P(α, β, γ) on the line; 87292.png is (x1 + , y1 + , z1 + ), where λ = 87298.png.




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