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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).

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

(2) Direction Cosine and direction ratio's of a line 
Important Important Formulas - 3D Geometry Formulas for JEE and NEET


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

same sign either +ve or –ve should be taken throughout. 

note that d.r's of a line joining  x1 , y1 , z1  and  x, y, z2 are proportional to x2 – x1  , y2 – y1 and z2 – z

(b) If θ is the angle between the two lines whose d.c's are l, m1 , n and l, m2 , n2 cosθ = l1 l2 + m1 m2+n1 n hence if lines are perpendicular then  l1 l2 + m1m2+ nn2  = 0

if lines are parallel then 

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

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

note that if three lines are coplanar then Important Important Formulas - 3D Geometry Formulas for JEE and NEET

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

(4)Projection of  join of 2 points on line with d.c's  l, m, n are  l (x2 – x1) + m(y2 – y1) + n(z– 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

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

(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 + d= 0  and  a2x + b2y + c2z + d2 = 0 are  perpendicular if a1 a2 + b1 b2 + c1 c2 = 0,

parallel if Important Important Formulas - 3D Geometry Formulas for JEE and NEET and    coincident if Important Important Formulas - 3D Geometry Formulas for JEE and NEET

(vi) Angle between a plane and a line is the compliment of the angle between the normal to the plane and the
Important Important Formulas - 3D Geometry Formulas for JEE and NEET

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

(viii) Distance between two parallel planes ax + by + cz + d1 = 0 and  ax+by + cz + d2 = 0  is Important Important Formulas - 3D Geometry Formulas for JEE and NEET

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

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

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 Pand Pis given by P1+Important Important Formulas - 3D Geometry Formulas for JEE and NEETP2=0 


C Straight Line in Space

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

Important Important Formulas - 3D Geometry Formulas for JEE and NEET and the lines through  (x1 , y1 ,z1) and (x2 , y2 ,z2).

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

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

 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.

The document Important Important Formulas - 3D Geometry Formulas for JEE and NEET is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on Important Important Formulas - 3D Geometry Formulas for JEE and NEET

1. What are the formulas for finding the distance between two points in 3D geometry?
Ans. The formula for finding the distance between two points (x1, y1, z1) and (x2, y2, z2) in 3D geometry is: Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
2. How do I calculate the midpoint between two points in 3D space?
Ans. To calculate the midpoint between two points (x1, y1, z1) and (x2, y2, z2) in 3D space, you can use the following formula: Midpoint = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)
3. What is the formula for finding the equation of a plane in 3D geometry?
Ans. The formula for finding the equation of a plane in 3D geometry, given three non-collinear points (x1, y1, z1), (x2, y2, z2), and (x3, y3, z3), is: Ax + By + Cz + D = 0 where A, B, C are the coefficients of the normal vector of the plane, and D is a constant term. These coefficients can be found using the cross product of two vectors formed by the three given points.
4. How can I find the angle between two vectors in 3D space?
Ans. To find the angle between two vectors in 3D space, you can use the dot product formula: Angle = acos((v1 · v2) / (|v1| |v2|)) where v1 and v2 are the two vectors, · represents the dot product, and |v1| and |v2| represent the magnitudes of the vectors.
5. How do I calculate the volume of a parallelepiped in 3D geometry?
Ans. The volume of a parallelepiped formed by three non-coplanar vectors a, b, and c can be calculated using the scalar triple product: Volume = |a · (b × c)| where × represents the cross product, · represents the dot product, and |a|, |b|, and |c| represent the magnitudes of the vectors.
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