Glossary
Conic Sections
Geometric shapes (circles, ellipses, parabolas, and hyperbolas) formed by the intersection of a plane with a double-napped cone.
Example:
The path of a comet around the sun can often be modeled as an ellipse, which is a type of Conic Section.
Direction (of a Vector)
The orientation of a vector in space, typically expressed as an angle relative to a reference axis.
Example:
A ship sailing due east has a Direction of 0 degrees or 90 degrees clockwise from north.
Implicitly Defined Functions
Functions where the relationship between variables (like x and y) is given by an equation that is not explicitly solved for one variable in terms of the other.
Example:
The equation x^2 + y^2 = 9 describes a circle and is an Implicitly Defined Function because y is not isolated.
Linear Transformations
Functions that map vectors from one vector space to another, preserving vector addition and scalar multiplication, often represented by matrix multiplication.
Example:
Rotating an image on a screen is a Linear Transformation that can be performed by multiplying the image's coordinate matrix by a rotation matrix.
Magnitude (of a Vector)
The length or size of a vector, calculated using the Pythagorean theorem for its components.
Example:
If a force vector is <3, 4> Newtons, its Magnitude is 5 Newtons, indicating the strength of the force.
Matrices
Rectangular arrays of numbers, symbols, or expressions arranged in rows and columns, used for organizing data and performing linear transformations.
Example:
A spreadsheet containing student grades for different subjects is conceptually similar to a Matrix.
Matrix Operations
Rules for combining or manipulating matrices, including addition, subtraction, multiplication, finding the inverse, and calculating the determinant.
Example:
When scaling an image in computer graphics, Matrix Operations like matrix multiplication are often used to transform the coordinates of each pixel.
Parameter
An independent variable, typically 't', used to define the coordinates (x, y) of a point on a curve in parametric equations.
Example:
In the equations x = 3cos(t) and y = 3sin(t) for a circle, 't' is the Parameter representing the angle.
Parametric Functions
Functions that define both x and y coordinates using a third independent variable, called a parameter, often 't' for time.
Example:
A drone's flight path can be described by Parametric Functions like x(t) = 5t and y(t) = -4.9t^2 + 20t, showing its horizontal and vertical position over time.
Parametrization (of Implicit Functions)
The process of expressing an implicitly defined function using parametric equations, often simplifying analysis or graphing.
Example:
To graph the circle x^2 + y^2 = 25 more easily, you can use Parametrization as x = 5cos(t) and y = 5sin(t).
Planar Motion
The movement of an object within a two-dimensional plane, often effectively modeled using parametric functions.
Example:
Tracking a car driving on a flat road is an example of Planar Motion, where its position (x, y) changes over time.
Rates of Change (Parametric)
Measures how the x and y coordinates of a parametric function change with respect to the parameter, typically expressed as dx/dt and dy/dt.
Example:
If a rocket's horizontal position is x(t) and vertical is y(t), then dx/dt tells you its horizontal speed, and dy/dt its vertical speed, representing its Rates of Change.
Vector Operations
Mathematical procedures like addition, subtraction, dot product, and cross product used to combine or manipulate vectors.
Example:
To find the resultant force of two forces acting on an object, you would use Vector Operations like vector addition.
Vector-Valued Functions
Functions that output a vector for each input value (often a scalar parameter like time), commonly used to describe paths or positions.
Example:
The position of a satellite orbiting Earth can be described by a Vector-Valued Function r(t) = <x(t), y(t), z(t)>.
Vectors
Mathematical objects possessing both magnitude (size or length) and direction, often represented graphically as arrows.
Example:
A strong wind blowing at 30 mph to the northeast can be represented as a Vector.