Q 9. - Calculus

Question

Explain why a line $\mathfrak{L}$and a point P not on $\mathfrak{L}$ determine a unique plane. Explain how you would use the equation of $\mathfrak{L}$ and the coordinates $P$ to find the equation of the plane. Explain why $P$ and $\mathfrak{L}$ do not determine a unique plane if $P$ is on $\mathfrak{L}$

Step-by-Step Solution

Answer

Because in ${\mathrm{ℝ}}^3$ there are infinitely many planes containing every line.

Step 1: Given information

A line $\mathfrak{L}$ and a point $P$ not on $\mathfrak{L}$

Step 2: Calculation

The goal is to show why it determines a unique plane and how to find the plane's equation.

A unique plane is determined by the three non-collinear points. Because the three points are non-collinear, any two unique points on $\mathfrak{L}$ as well as the point $P$ will determine a plane.

The following steps are followed to find the equation of the plane from a point $P$ not on line $\mathfrak{L}$ and the line $\mathfrak{L}$

• Choose any point $Q$ on the line $\mathfrak{S}$
• Construct the vector $\overrightarrow{PQ}$
• Find the normal vector $\mathbf{N}$ by $\mathbf{N}=\overrightarrow{PQ}\times \mathbf{d}$, where $\mathbf{d}$ is the direction vector of the line $\mathcal{L}$
• Use $P$ and $\mathbf{N}$ to find the equation of the plane.

Step 3: Calculation

The point $P$ and the line $\mathfrak{L}$ don't determine a unique plane if the point $P$ is on the line $\mathfrak{L}$ because in${\mathrm{ℝ}}^3$ there are infinitely many planes containing every line.