# Equation of Lines in Space

### Vector Form

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

### Cartesian Equations

A line can be determined by the 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 .

#### Cartesian Equations

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

3.Given the line r:

Find its equations in parametric and Cartesian form.

4The equation of a line r is:

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

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).

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.