Cartesian Equations of a Line

Given a point, P = (x0, y0, z0), and a directional vector u, the Cartesian equation of a line is:

Cartesian Equations of the Line

Problems

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

Cartesian Equation Operations

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

Cartesian Equations


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

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





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