Linear programming is used to solve an optimization problem wherein the objective function is the liner function which is essentially referred to as the optimization equation. Discrete optimization is a branch of optimization methodology containing discrete quantities which are non-continuous functions. It is quite ubiquitous in as diverse applications such as financial investment, diet planning, … Read more
Decision variables are used as mathematical symbols representing levels of activity of a firm. Parameters are the numerical coefficients and constants used in the objective function and constraint equations. Likewise, constraints are restrictions placed on the firm by the operating environment stated in linear relationships of the decision variables. Constraints illustrate all the possible values … Read more
A mathematical model is a set of equations and inequalities that describe a system. Eg.E = mc2, Y = 5.4 + 2.6 X. These models are designed to have several types of programming: Linear programming Integer programming Non- linear programming A linear programming procedure involves mathematical technique to solve optimization models with linear objectives and … Read more
Consider that we are given a point Q, not in a plane and a point P on the plane and our goal for the question is to find the shortest distance possible between the point Q and the plane. The shortest distance between any two points is at a perpendicular state. The distance between a … Read more
The angle between a line, d, and a plane, π, is the angle between d and its orthogonal projection onto π, d’. Accordingly, the angle between a line and a plane can be considered equivalent to the complementary acute angle that forms between the direction vector of the line $\overrightarrow{d}=(a_{2}, b_{2},c_{2})$ and the normal vector, … Read more
The angle between two planes is considered the same as the angle between their 2 normal vectors as per the geometrical arrangement shown below. These lines are perpendicular to the line of intersection of the planes. In the figure above, there is a quadrilateral that has 2 angles as shown adding to 180°. So θ … Read more
Consider two vectors: then the acute angle θ between two straight lines is given by: Where b and d can be considered as the direction vectors of the two lines as shown above. The lines do not have to be intersecting – the angle is the angle between them if one was moved along so … Read more
A plane can be completely illustrated by denoting two intersecting lines which can be translated into a fixed point A and two nonparallel direction vectors. The position vector $\overrightarrow{r}$ of any general point P on the plane passing through point A and having direction vectors $\overrightarrow{b}$ and $\overrightarrow{c}$ is given by the equation Vector equation … Read more
Cartesian equation and vector equation of a line, coplanar and skew lines, the shortest distance between two lines The vector $\overrightarrow{AB}$ has a definite length while the line AB is a line passing through the points A and B and has infinite length. It can be identified by a linear combination of a position vector … Read more
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