Q. 3 TF - Calculus

Question

Give two other sets of geometric conditions that would uniquely determine a plane in ${\mathrm{ℝ}}^3$.

Step-by-Step Solution

Answer

Three non-collinear points determine a plane ${\mathrm{ℝ}}^3$.

A plane in ${\mathrm{ℝ}}^3$ is determined by a point on the plane and two direction vectors. 

Step 1. Given Information

Give two other sets of geometric conditions that would uniquely determine a plane in ${\mathrm{ℝ}}^3$.

Step 2. Three non-collinear points determine a plane.

This statement indicates that if three points are not on the same line, only one plane can pass between them. Because the points show you exactly where the plane is, the plane is determined by the three points.

Step 3. A plane in ℝ 3 is determined by a point on the plane and two direction vectors.

A plane is defined by a point on the plane and two direction vectors running parallel to it. The reason that we require two parallel vectors parallel to the plane instead of one for the line indicates that the plane is two dimensional while the line is one.