Q 15. - Calculus

Question

Given the equations $x=at+x_0,y=bt+y_0,z=ct+z_0$ for a line $\mathfrak{L}$ and $\alpha x+\beta y+\gamma z=\delta$ for a plane $\rho$ explain how to determine whether $\mathfrak{L}$ is orthogonal to $\rho$

Step-by-Step Solution

Answer

The line and the plane are orthogonal if and only if $\mathbf{d}=⟨a,b,c⟩$ and $\mathbf{N}=⟨\alpha ,\beta ,\gamma ⟩$ are scalar multiple of each other.

Step 1: Given information

The equations $x=at+x_0,y=bt+y_0,z=ct+z_0$ for a line $\mathfrak{L}$ and $\alpha x+\beta y+\gamma z=\delta$ for a plane $\wp$

Step 2: Calculation

The goal is to show how to figure out whether a line $\mathfrak{L}$ is orthogonal to a plane $\rho$ Find the direction vector and the normal vectors for the given line and plane.

Step 3: Calculation

The direction vector for the line $x=at+x_0,y=bt+y_0,z=ct+z_0$ is $\mathbf{d}=⟨a,b,c⟩$

The normal vector of the plane $\alpha x+\beta y+\gamma z=\delta$ is $\mathbf{N}=⟨\alpha ,\beta ,\gamma ⟩$

The line and the plane are orthogonal if and only if $\mathbf{d}=⟨a,b,c⟩$ and $\mathbf{N}=⟨\alpha ,\beta ,\gamma ⟩$ are scalar multiple of each other.