Pair of Straight Lines – Theorem Combined Equation of Pair of Lines joining Origin and the Point of Intersection of a Curve and A Line Let us find the equation of the straight line joining the origin and the Points of intersection of the curve ax² + 2hxy + by² + 2gx + 2fy + Read more about Pair of Straight Lines – Theorem[…]
Polar Coordinates Polar Coordinates express the location of a point as (r, θ). Where r is the distance from the origin (pole) to the point and θ is the angle from the positive x – axis to the point (in degree or radius). Here, r and θ are called polar coordinates. The distance r is Read more about Polar Coordinates[…]
Shortest Distance between Two Skew Lines Definition: l₁ and l₂ are two skew lines. If P is a point on l₁ and Q is a point on l₂ such that PQ perpendicular to l₁ and PQ is called shortest distance and PQ is called shortest distance line between the lines l₁ and l₂. Theorem: The Read more about Shortest Distance between Two Skew Lines[…]
Vector Triple Product The vector triple product of three vectors , and is the vector . Also, . In general, . If , then the vectors and are collinear. is a vector perpendicular to and but is a vector perpendicular to the plane of and . Hence, vector must lie in the plane of and Read more about Vector Triple Product[…]
Applications of Dot (Scalar) Product Finding Angle between Two Vectors If and are non-zero vectors, then the angle between them is given by . Also . Or . Cosine Rule using Dot Product: Using vector method, prove that in a triangle a² = b² + c² – 2bc cos A (Cosine law). In ΔABC, Let Read more about Applications of Dot (Scalar) Product[…]
Properties of Vector Addition 1. Communitive property . 2. Associative property . 3. Additive identity . 4. Additive inverse . 5. and . Examples 1: If vector bisects the angle between and , then prove that . Solution: We know that vector is along the diagonal of the parallelogram whose adjacent sides are vectors and Read more about Properties of Vector Addition[…]
Centroid of a Tetrahedron Tetrahedron: Let ABC be a triangle and D is a point in the space which is not in the plane of the triangle ABC. Then ABCD is called a tetrahedron. Note: The tetrahedron ABCD has four faces, namely ∆ABC, ∆ACD, ∆ABD, ∆BCD. It has four vertices, namely A, B, C, D Read more about Centroid of a Tetrahedron[…]
Centroid of a Triangle Centroid: The centroid of the triangle formed by the points (x₁, y₁, z₁), (x₂, y₂, z₂), (x₃, y₃, z₃) is . Proof: Let A (x₁, y₁, z₁), B (x₂, y₂, z₂) and C (x₃, y₃, z₃) Let D be the midpoint of BC. Then . Since centroid G divides each median Read more about Centroid of a Triangle[…]
Bernoulli’s Equation Solution of Bernoulli’s Equation: Given equation is dy/ dx + Py = Qyⁿ 1/ yⁿ dy/ dx + P. 1/ yⁿ¯¹ = Q. , then , Becomes , , It is linear in v and can be solved. Example: Solve . Solution: Given that , , , , … (1) Put tany = Read more about Bernoulli’s Equation[…]
Linear Differential Equation Definition: An equation of the form dy/dx + Py = Q, where P and Q are function of x only, is called a Linear differential equation. Solution of Linear Equation: Given equation is dy/dx + P y = Q , , , . Example 1: Solve (1 + x²) dy/dx + y Read more about Linear Differential Equation[…]