Foot of perpendicular from a point on the given Line: Here, the equation of line AB is . Let L be the foot of the perpendicular from P (α, β, γ) on the line . Let the coordinates of L be (x₁ + aλ, y₁ + bλ, z₁ + cλ). Then the direction ratios of Read more about Perpendicular Distance of a Point from a Line – Cartesian form[…]
From the figure, and . Since is collinear with , for some scalar λ, we have . or . or … (i) Therefore, the equation of a straight line passing through and is . We have and . Substituting these values in (i), we get . Equating the coefficients of , and , we get Read more about Equation of Line passing through Two given Points[…]
Equation of Straight Line passing through a given Point and Parallel to a given Vector Vector Form: Line passing through point and parallel to vector . Let A be the given point and let AP be the given line through A. Let be any vector parallel to the given line. Position vector of point A Read more about Vectors – Equation of Straight Line[…]
Homogeneous Differential Equation A differential equation is said to be a homogeneous differential equation in x, y if both f (x, y), g (x, y) are homogeneous function of same degree in x and y. To find the solution of the Homogeneous Differential Equation put y = v x, then . Substituting these values in Read more about Homogeneous Differential Equation[…]
Complex Numbers – Equation of a Circle Equation of a Circle: Consider a fixed complex number zₒ and let z be any complex number which moves in such a way that its distance from zₒ is always to r. this implies z would lie on a circle whose center is zₒ and radius is r. Read more about Complex Numbers – Equation of a Circle[…]
Remainder and Factor Theorems Remainder Theorem: The remainder theorem states that if a polynomial f(x) is divided by a linear function x – k, then the remainder is f(k). Proof: In any division Divided = divisor x Quotient + Remainder Let Q(x) be the quotient and R be the remainder Then f (x) = (x Read more about Remainder and Factor Theorems[…]
Area of a Triangle The area of a triangle, whose vertices are (x₁, y₁), (x₂, y₂), and (x₃, y₃) is = ½ |x₁ (y₂ – y₃) + x₂ (y₃ – y₁) + x₃ (y₁ – y₂)|. Proof: Let ABC be a triangle whose vertices are A (x₁, y₁), B (x₂, y₂), and C (x₃, y₃). Read more about Area of a Triangle[…]
Vector Along the Bisector of Given Two Vectors We know that the diagonal in a parallelogram is not necessarily the bisector of the angle formed by two adjacent sides. However, the diagonal in a rhombus bisects the angle between the two adjacent sides. Consider vectors and forming a parallelogram ABCD as shown in figure. Consider Read more about Vector Along the Bisector of Given Two Vectors[…]
Vector Joining Two Points Vector is shifted without rotation and placed at origin. Now vector . Since , coordinates of point C are (x₂ – x₁, y₂ – y₁, z₂ – z₁) Hence, Vector , , . = Position vector of B – Position vector of A Also, from above, we have which describes triangle Read more about Vector Joining Two Points[…]
Multiplication of a Vector by a Scalar Let be a vector and λ a scalar. Then the product of vector by scalar λ, denoted by , is called the multiplication of vector by the scalar λ. Note that is also a vector, collinear to vector . Vector has the direction same (or opposite) as that Read more about Multiplication of a Vector by a Scalar[…]