Q 10. - Calculus

Question

Explain why two intersecting lines determine a unique plane. Explain how you would use the equations of the lines to find the equation of the plane.

Step-by-Step Solution

Answer

The two intersecting lines determine a unique plane.

Step 1: Given information

The two intersecting lines.

Step 2: Calculation

The goal is to explain why two intersecting lines produce a single plane.

The lines cross at one place and intersect at another. In some planes, the point of intersection is located.

Assume that each line has two more points. The three points will not be parallel. A unique plane is always determined by the non-collinear points.

Therefore, the two intersecting lines determine a unique plane.

Step 3: Calculation

The steps for determining a planar equation from two intersecting lines are as follows:

• First find the direction vectors ${\mathbf{d}}_1$ and ${\mathbf{d}}_2$ of both lines ${\mathbf{r}}_1\left(t\right)$ and ${\mathbf{r}}_2\left(t\right)$
• Find the normal vector $\mathbf{N}=⟨a,b,c⟩$ using the relation $\mathbf{N}={\mathbf{d}}_1\times {\mathbf{d}}_2$
• Find the point of intersection $(\alpha, \beta, \gamma)$ by equating the direction vectors ${\mathbf{d}}_1$ and ${\mathbf{d}}_2$
• The equation of the plane is given by $a(x-\alpha )+b(y-\beta )+c(z-\gamma )=0$