Equation of a Line in Space (Vector and Cartesian) A line in space can be defined mathematically in two ways: 1. Vector Equation: A line in space can be...
Equation of a Line in Space (Vector and Cartesian)
A line in space can be defined mathematically in two ways:
1. Vector Equation:
A line in space can be represented by a vector equation of the form:
where:
(\vec{r}) is the position vector, a vector with three components representing the coordinates of a point on the line
(x, y, z) are the coordinates of the point
2. Cartesian Equation:
Another way to represent a line in space is through a Cartesian equation of the form:
where:
(A, B, C) are constants representing the coefficients of the variables in the equation
(D) is the constant on the right side
Examples:
1. Vector Equation:
This vector equation represents the line passing through the points ((-2, 3, 4)) and ((6, 9, 12)).
2. Cartesian Equation:
This Cartesian equation represents the line passing through the points ((-1, 2, 3)) and ((4, 7, 1)).
Key Points:
Both vector and Cartesian equations provide the same information about a line in space.
The vector equation is more general, while the Cartesian equation is more specific.
The vector equation is often used when the line is defined by a set of points, while the Cartesian equation is often used when the line is defined by an equation