Let a line AB in 3D space make angles α, β, γ respectively with the +ve direction of coordinate axes X, Y, Z. Therefore, we express cosα, cosβ, cosγ as direction cosines of the line AB in the 3D space. Clearly; direction cosines fix the direction cosines of a line in space. Also, parallel lines … Read more
Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, a scalar triple product of vectors The dot/scalar product of two vectors $\overrightarrow{a}$and $\overrightarrow{b}$ is:$\overrightarrow{a}. \overrightarrow{b}= \overrightarrow{b} . \overrightarrow{a} =\vert \overrightarrow{a}\vert \vert \overrightarrow{b}\vert \cos \theta $ It is essentially the product of the length of one of them and projection of the other … Read more
Equal vectors: Vectors with the same length and direction, and must represent the same quantity (such as force or velocity). Unit vector: For this vector, the length is always 1.For a vector 𝑎 ⃗ , a unit vector is in the the same direction as 𝑎 ⃗ and is given by: ${a} = \frac{\overrightarrow{a}}{\vert \overrightarrow{a}\vert }$ $\begin{matrix} a_{x}=(x_{B}-x_{A}) & … Read more
Consider a vector as shown below on the x-y-z plane. The angles made by this line with the +ve direactions of the coordinate axes: θx, θy, and θz are used to find the direction cosines of the line: cos θx, cos θy, and cos θz. Likewise, the direction cosine of θx, θy, and θz in … Read more
Many quantities in geometry and physics, such as area, time, and temperature are presented using a single real number. Other physical and algebraic quantities, such as velocity and momentum, have both magnitude and direction and cannot be completely characterized by a single real number. A scalar quantity can be described by a single number characterized … Read more
Integration of a variety of functions by substitution, by partial fractions and by parts Integration by substitution Consider the function: To find the integral of a complex function such as this, we introduce a new variable. We change the function of x from the variable x to a new variable u.Suppose that we let u … Read more
Solution for dx/dy + px = q, where p and q are functions of y or constants A differential equation is called linear if there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables. It can be written in two forms. The second is discussed … Read more
Solutions of the linear differential equation of the type − dy/dx + py = q, where p and q are functions of x or constants A differential equation is called linear if there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables. It can be written … Read more
The solution of differential equations by the method of separation of variables solutions of homogeneous differential equations of the first order and first degree Method of separation of variables is one of the most widely used techniques to solve ODE. It is based on the assumption that the solution of the equation is separable. This … Read more
A solution of a differential equation is an expression to show the dependent variable in terms of the independent one(s) I order to satisfies the relation between variables. The general solution includes all possible families of solutions and it includes arbitrary constants (in the case of an ODE) or arbitrary functions (in the case of … Read more