Important Important Formulas - 3D Geometry Formulas for JEE and NEET

3 -D Coordinate Geometry

(1) Distance (d) between two points  (x₁ , y₁ , z₁) and (x₂ , y₂ , z₂).

     =

(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  x₁ , y₁ , z₁  and  x₂ , y₂ , z₂ are proportional to x₂ – x₁  , y₂ – y₁ and z₂ – z₁ 

(b) If θ is the angle between the two lines whose d.c's are l₁ , m₁ , n₁  and l₂ , m₂ , n₂ cosθ = l₁ l₂ + m₁ m₂+n₁ n₂  hence if lines are perpendicular then  l₁ l₂ + m₁m₂+ n₁ n₂  = 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 (x₂ – x₁) + m(y₂ – y₁) + n(z₂ – z₁) 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 (x₁ , y₁ , z₁) is a (x – x₁) + b (y – y₁) + c (z – z₁) = 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 co-ordinate 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   a₁ x + b₁ y + c₁z + d₁ = 0  and  a₂x + b₂y + c₂z + d₂ = 0 are  perpendicular if a₁ a₂ + b₁ b₂ + c₁ c₂ = 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 (x₁ , y₁ , z₁) to a plane ax + by + cz + d = 0 is

(viii) Distance between two parallel planes ax + by + cz + d1 = 0 and  ax+by + cz + d₂ = 0  is

(ix) Planes bisecting the angle between two planes a₁x + b₁y + c₁z + d1 = 0  and  a₂ + b₂y + c₂z + d₂ = 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 P₁ and P₂ is given by P₁+P₂=0 

C Straight Line in Space

(i) Equation of a line through A (x₁ , y₁ , z₁) and having direction cosines  l ,m , n  are 

 and the lines through  (x₁ , y₁ ,z₁) and (x₂ , y₂ ,z₂).

(ii) Intersection  of two planes  a₁x + b₁y + c₁z + d1 = 0   and   a₂x + b₂y + c₂z + d₂ = 0 together represent the unsymmetrical form of the straight line.

(iii) General equation of the plane containing the line  

 is A (x – x₁) + B(y – y₁) + c (z – z₁) = 0    where  Al + bm + cn = 0 .

Line of greatest slope
AB is the line of intersection of G-plane 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.