Inequalities Involving A.M, G.M, H.M Let A, G and H be Arithmetic, Geometric, and Harmonic means of two positive numbers a and b. Then, , and . These three means possess the following property. A ≥ G ≥ H. Proof: We have , and . , , , , A ≥ G … (1) , Read more about Inequalities Involving A.M, G.M, H.M[…]
Diameter from of the Equation of a Sphere Let AB be the diameter of a sphere whose centre is C. let the vectors of the extremities A and B of the diameter be and , respectively. Let P be any point on the spheres. Suppose the position vector of P is . We know that Read more about Diameter from of the Equation of a Sphere[…]
Regular Tetrahedron A regular tetrahedron is one in which all four faces are equilateral triangles. Angle between any Edge and Face not Containing that Edge of Regular Tetrahedron: Let us find the angle between edge OA and face OBC. Since tetrahedron is regular and its face is an equilateral triangle, the required angle is the Read more about Regular Tetrahedron[…]
Image of a Point in a Plane Distance of a point From a Plane – Cartesian Form: Let PM be the length of the perpendicular from a point P (x₁, y₁, z₁) to the plane ax + by + cz + d = 0. Then the equation of PM is . The coordinates of any Read more about Image of a Point in a Plane[…]
Distance of a Point from a Plane – Vector Form Vector Form: Let be the given plane and be the given point. Let PM be the length of the perpendicular from P to the Plane π. Since line P passes through and is parallel to vector , which is normal to the Plane π, the Read more about Distance of a Point from a Plane – Vector Form[…]
Equation of a Plane Passing Through the Line of Intersection of Two Planes Let π₁ and π₂ be two planes with equations and , respectively. The position vector of any point on the line of intersection must satisfy both the equation. If is the position vector of a point on the line, then and . Read more about Equation of a Plane Passing Through the Line of Intersection of Two Planes[…]
Equation of the Plane that Passes through Point A with Position Vector and is Parallel to given Vector and : Vector form: Let be the position vector of any Point P in the plane. Then , Since vectors , and are coplanar, we have , , , , Which is the required equation of the Read more about Vector – Equation of a Plane[…]
Equation of a Plane Passing through Two Parallel Lines Let the plane pass through parallel lines and . As shown in the diagram, for any position of R in the plane, vectors , and are coplanar. Then , which is the required equation of the plane. Equation of Plane Parallel to a Given Plane: The Read more about Equation of a Plane Passing through Two Parallel Lines[…]
Equation of a Plane in Normal Form and Cartesian Form Plane: A plane is a surface such that if any two points are taken on it, the line segment joining them lies completely on the surface. A plane is determined uniquely if. The normal to the plane and its distance from the origin is given, Read more about Equation of a Plane in Normal Form and Cartesian Form[…]
Let L be the foot of the perpendicular draw from on the line . Since denotes the position vector of any point on the line , the position vector of L will be . Direction ratios of PL = . Since is perpendicular to , we have . . . Thus, position vector of L Read more about Perpendicular Distance of a Point from a Line – Vector Form[…]