The angle between two planes is the angle formed between their normal vectors. To find the normal vectors of two planes, you can use the cross product of their respective direction vectors. Let’s say we have two planes represented by their equations: Plane 1: ax + by + cz + d1 = 0 Plane 2:…
The angle between two lines can be found using the slope of each line. If the slopes of the lines are m1 and m2, then the angle between the lines is given by the formula: θ = arctan(|(m1 – m2)/(1 + m1*m2)|) where arctan is the inverse tangent function. Note that the absolute value is…
Direction cosines and direction ratios are used to describe the orientation of a line or a vector in three-dimensional space. Direction cosines are the cosines of the angles that a given line or vector makes with the positive x, y, and z axes of a Cartesian coordinate system. For example, if the angles that a…
An ellipse is a geometric shape that looks like a flattened circle. It is defined as the set of all points in a plane, the sum of whose distances from two fixed points (called the foci) is constant. The standard form equation of an ellipse centered at the origin is: (x^2 / a^2) + (y^2…
Trigonometry is a branch of mathematics that deals with the study of relationships involving angles and the sides of triangles. It has many practical applications in fields such as engineering, physics, and astronomy. One of the most important aspects of trigonometry is the periodicity of the trigonometric functions. The six trigonometric functions are sine, cosine,…
Matrices are often used to solve systems of simultaneous linear equations in two or three variables. The matrix method involves representing the coefficients of the equations and the constants as matrices, and then performing matrix operations to obtain the solution. Let’s take an example of two linear equations in two variables: 3x + 2y =…
