Equation of Lines in Space

Vector Form


Equation of a Line

If P(x1, y1, z1) is a point on the line r and the vector Vector PX has the same direction as Vector U, then it is equal to Vector U multiplied by a scalar:

Line in Space

Vector Line Equation

Vector Line Equation


Parametric Form

Parametric Form of the Line

Parametric Form of the Line

Cartesian Equations

Cartesian Equations of the Line


A line can be determined by the intersection of two planes.

Intersection of Two Planes


Examples

1.Find the equations of the line that pass through the point A = (1, 2, 1) and whose direction vector is vector.

Parametric Form

Parametric Form Example

Cartesian Equations

Cartesian Equation Example

Cartesian Equation Operations

Cartesian Equations


2.Find the equations of the line that pass through the points A = (1, 0, 1) and B = (0, 1, 1).

Lines in Space Example

Parametric Form Example

Cartesian Equations

Cartesian Equations

Cartesian Form Solution


3.Given the line r:

Equation of the Line

Find its equations in parametric and Cartesian form.

Equation of Lines Example

Line Equation Calculations

Line Equation Calculations

Line Equation Calculations

Line Equation Operations

Line Equation Solutions

Solutions


4The equation of a line r is:

Equation of a Line Example

Do the points A = (0, −2, −2) and B = (3, 2, 6) belong to the line r?

Line Equation Example

Line Equation Example


5.Find the equation of the line that is parallel to the lines given by x = 3λ, y = λ and z = 2λ + 2, and contains the point P(0, 1, −1).

Line Equation Example

Equation Solution


6.A straight line is parallel to the planes x + y = 0 and x + z = 0. If it passes through the point (2, 0, 0), find its equation.

The direction vector of the line is perpendicular to the normal vectors of each plane.

Normal Vectors

Cross Product

Line Equation Example

Line Solution