Vector and Cartesian Equations of a Line in Space A line in space can be defined by both a vector equation and a Cartesian equation. Vector Equation: A...
Vector and Cartesian Equations of a Line in Space
A line in space can be defined by both a vector equation and a Cartesian equation.
Vector Equation:
A vector equation for a line in space can be written in the form:
r(t) = <a, b, c> + t<d, e, f>
where:
r(t) is a point on the line at time t
a, b, c, d, e, f are constants
Cartesian Equation:
A Cartesian equation for a line in space can be written in the form:
Ax + By + Cz = D
where:
A, B, C are constants
D is a constant
Both vector and Cartesian equations describe the same line.
Examples:
Vector Equation:
r(t) = <1, 2, 3> + t<4, 5, 6>
Cartesian Equation:
3x - 2y + z = 1
Key Points:
A line in space can be defined by both a vector equation and a Cartesian equation.
A vector equation describes the line as a set of points that move in a straight line with a constant speed, while a Cartesian equation describes the line as a set of points that lie on a plane with a constant slope.
Both vector and Cartesian equations can be used to find the equation of a line in space