**A line and a plane**

A line in space can be expressed using vectors. When the point and the direction vector of a line are known, the line is expressed by the vectors

where *s* is the name of the line, *OA* is the position vector of the line, *a* is the direction vector of the line, and n is some number.

**Example 1**

A line passes through space through points* A (1,3,2)* and* B (2,4,1)*. Find the vector equation of the line. The direction vector of the line is the vector AB.

The vector equation of the line is

**Example 2**

Is the point (4,6, -1) on the line of the previous example?

A parametric representation of the line* s* is formed

We get a system of equations

The point is on the line only and only if the number *n* is found such, that the system of equations is true. Substitute the point (4,6, -1) into the system of equations.

The system of equations is true when n = 3. So the point is on the line.

A plane is formed by three points that are not on the same line. Plane direction vectors are obtained from point-to-point vectors.

**Example 3**

A plane passes through points *A (−2,2,2), B (1,1,0)* and *C (2,1,2)*. Under what condition is point *P (x, y, z)* on the plane?

We define a vector representation of the plane.

Position vectors and direction vectors

The vector representation of the plane

We get a system of equations

where *n* and* t* are some numbers. The point is on the plane only and only if the numbers *n* and* t *are found such that the system of equations is true.

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