Coplanar

Two or more vectors are coplanar if they are linearly dependent, therefore their components are proportional and the rank is 2.

Two or more points are coplanar if the vectors determined by them are also coplanar.

Examples

1. Determine if the points A = (1, 2, 3), B = (4, ,7, 8), C = (3, 5, 5), D = (−1, −2, −3) and E = (2, 2, 2) are coplanar.

The points A, B, C, D and E are coplanar if:

Vector Rank

Coplanar Vector Example

Coplanar Vector Example

Coplanar Vector Example

Coplanar Vector Example

Matrix

Vector Rank

The points A, B, C, D and E are not coplanar.


2.Calculate the value of x for the coplanar set of points A = (0, 0, 1), B = (0, 1, 2), C = (−2, 1, 3) and D = (x, x−1, 2).

Coplanar Vector Example

Coplanar Vector Example

Coplanar Vector Example

Coplanar Vector Calculations

Coplanar Vector Solution


3.What are the values of a, b and c so that the points A = (1, 0, 1), B = (1, 1, 0), C = (0, 1, 1) and D = (a, b, c) are coplanar?

The points A, B, C and D are coplanar if:

Vector Rank

Coplanar Vector Example

Coplanar Vector Example

Coplanar Vector Example

Matrix

Coplanar Vector Solution


4.Calculate the value of a for the points (a, 0, 1), (0, 1, 2), (1, 2, 3) and (7, 2, 1) so that they are coplanar. Also, calculate the equation of the plane that contains them.

Vector Rank

Coplanar Vector Example

Coplanar Vector Example

Coplanar Vector Example

Coplanar Vector Calculations

Linear Determination

Equation of the Plane