Q 8. - Calculus

Question

Explain why three noncollinear points determine a unique plane. Explain how you would use the coordinates of the points to find the equation of the plane. Explain why three collinear points do not determine a unique plane.

Step-by-Step Solution

Answer

• The three non-collinear points determine a unique plane.
• Collinear points don't determine a unique plane.

Step 1: Given information

Consider three non-collinear points.

Step 2: Calculation

The goal is to demonstrate why it determines a unique plane and how to obtain the plane's equation from three non-collinear locations.

The three non-collinear points serve as the triangle's vertices, which are utilized to draw the plane containing the points.

As a result, a unique plane is determined by the three non-collinear points.

To obtain the equation of the plane from three non-collinear locations $P, Q$ and $R$ follow the procedures below.

• Construct the vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$
• Find the normal vector $\mathbf{N}=⟨a,b,c⟩$ by the relation $\mathbf{N}=\overrightarrow{PQ}\times \overrightarrow{PR}$
• Use $\mathbf{N}=⟨a,b,c⟩$ and point $P$  to find the equation of the plane.

Step 3: Calculation

Because three collinear points identify an infinite number of planes, they do not determine a single plane. Several aircraft can pass through the same line at the same time. As a result, collinear points do not determine the existence of a single plane.