3 D Coordinate Geometry
(1) Distance (d) between two points (x_{1} , y_{1} , z_{1}) and (x_{2} , y_{2} , z_{2}).
=
(2) Direction Cosine and direction ratio's of a line
(3) Direction cosine of a line has the same meaning as d.c's of a vector.
(a) Any three numbers a, b, c proportional to the direction cosines are called the direction ratio s i.e.
same sign either +ve or –ve should be taken throughout.
note that d.r's of a line joining x_{1} , y_{1} , z_{1} and x_{2 }, y_{2 }, z_{2} are proportional to x_{2} – x_{1} , y_{2} – y_{1} and z_{2} – z_{1 }
(b) If θ is the angle between the two lines whose d.c's are l_{1 }, m_{1} , n_{1 } and l_{2 }, m_{2} , n_{2} cosθ = l_{1} l_{2} + m_{1} m_{2}+n_{1} n_{2 } hence if lines are perpendicular then l_{1} l_{2} + m_{1}m_{2}+ n_{1 }n_{2} = 0
if lines are parallel then
note that if three lines are coplanar then
(4)Projection of join of 2 points on line with d.c's l, m, n are l (x_{2} – x_{1}) + m(y_{2} – y_{1}) + n(z_{2 }– z_{1}) B PLANE
(i) General equation of degree one in x, y, z i.e. ax + by + cz + d = 0 represents a plane. (ii) Equation of a plane passing through (x_{1} , y_{1} , z_{1}) is a (x – x_{1}) + b (y – y_{1}) + c (z – z_{1}) = 0 where a, b, c are the direction ratios of the normal to the plane.
(iii) Equation of a plane if its intercepts on the coordinate axes are x1 , y1 , z1 is z 1 z
(iv) Equation of a plane if the length of the perpendicular from the origin on the plane is p and d.c's of the perpendicular as l , m, , n is l x + m y + n z = p
(v) Parallel and perpendicular planes – Two planes a_{1} x + b_{1} y + c_{1}z + d_{1 }= 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 are perpendicular if a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0,
parallel if and coincident if
(vi) Angle between a plane and a line is the compliment of the angle between the normal to the plane and the
where θ is the angle between the line and normal to the plane.
(vii) Length of the perpendicular from a point (x_{1} , y_{1} , z_{1}) to a plane ax + by + cz + d = 0 is
(viii) Distance between two parallel planes ax + by + cz + d1 = 0 and ax+by + cz + d_{2} = 0 is
(ix) Planes bisecting the angle between two planes a_{1}x + b_{1}y + c_{1}z + d1 = 0 and a_{2} + b_{2}y + c_{2}z + d_{2} = 0 is
Given by
Of these two bisecting planes , one bisects the acute and the other obtuse angle between the given planes.
(x) Equation of a plane through the intersection of two planes P_{1 }and P_{2 }is given by P_{1}+P_{2}=0
C Straight Line in Space
(i) Equation of a line through A (x_{1} , y_{1 }, z_{1}) and having direction cosines l ,m , n are
and the lines through (x_{1} , y_{1} ,z_{1}) and (x_{2} , y_{2} ,z_{2}).
(ii) Intersection of two planes a_{1}x + b_{1}y + c_{1}z + d1 = 0 and a_{2}x + b_{2}y + c_{2}z + d_{2} = 0 together represent the unsymmetrical form of the straight line.
(iii) General equation of the plane containing the line
is A (x – x_{1}) + B(y – y_{1}) + c (z – z_{1}) = 0 where Al + bm + cn = 0 .
Line of greatest slope
AB is the line of intersection of Gplane and H is the horizontal plane. Line of greatest slope on a given plane, drawn through a given point on the plane, is the line through the point 'P' perpendicular to the line of intersetion of the given plane with any horizontal plane.
209 videos443 docs143 tests

1. What are the formulas for finding the distance between two points in 3D geometry? 
2. How do I calculate the midpoint between two points in 3D space? 
3. What is the formula for finding the equation of a plane in 3D geometry? 
4. How can I find the angle between two vectors in 3D space? 
5. How do I calculate the volume of a parallelepiped in 3D geometry? 
209 videos443 docs143 tests


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