Q 17. - Calculus

Question

When a line $\mathfrak{L}$ intersects a plane $\rho$ the angle between them is defined to be the complement of the acute angle between the direction vector for the line and the normal vector to the plane. Draw a figure that illustrates this angle,

and explain why the definition given is a reasonable definition.

Step-by-Step Solution

Answer

The angle described is the minimum angle that the line $\mathfrak{L}$ can form with the line in the plane $\rho$ through $Q$

Step 1: Given information

A line $\mathfrak{L}$ intersecting the plane $\wp$

Step 2: Calculation

The purpose is to illustrate why the complement of the acute angle produced between the line's direction vector and the plane's normal vector is a fair definition of the angle generated when a line encounters a plane.

Assume that $Q$ is the point of intersection of the line and the plane.

Step 3: Calculation

If the line $\mathfrak{L}$ is orthogonal to the plane $\rho$ then the angle between them is $\frac{\pi }{2}$

Otherwise, the angle described is the minimum angle that the line $\mathfrak{L}$ can form with the line in the plane $\rho$ through $Q$