Yashavantrao Chavan Institute of Science Satara. Rayat Shikshan Sanstha’s Rayat Gurukul CET Project Std : XII Sub : -Mathematics

Topic :-Three Dimensional Geometry

 Class Exercise  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

O Y Z X A P

O Y Z X A B P

Find the vector and the Cartesian equations for the line through the points A(3, 4, -7) and B(5, 1, 6).

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 Let P be the point where the line AB crosses the y z-plane. Then, the position vector of the point P is

This point must satisfy (i) Solving (ii), (iii) and (iv), we get

If the lines intersect, then they have a common point.

P(1, 2, -3)

Direction ratios of the given line are 2, - 2, -1.

Q A B P (2, -1, 5) L 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.

Hence, the image of P(2, -1, 5) in the given line is (0, 5, 1).

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)

Solution: The lines will intersect if shortest distance between them = 0 Therefore, the given lines intersect.

Thank you