# Angle between line and its shadow on a plane

1. Jun 8, 2012

### manderson010

1. The problem statement, all variables and given/known data
A line intersects a vertical wall at 40.78° (that is, 49.22° with respect to the normal vector of the wall). The line is contained within a vertical plane perpendicular to the wall such that the aforementioned angle is measured with respect to the plumb line dropped from the point of intersection. The line casts a shadow on the wall that is 68.04° from the plumb line. What is the angle between the line and its shadow?

2. Relevant equations
I haven't the foggiest. There's got to be a simple expression out there somewhere for this, but I've never encountered it before.

Thank you very much for your help!

Personally, I would be inclined to use "vectors" to do this. Setting up a coordinate system so that the z-axis is up the wall is the yz-plane, and the line is in the xz-plane, we have a vector, of length 1, pointing along the line, given by sin(40.78)i+ cos(40.78)k. The shadow, making an ange of 68.04 with the z- axis, would be along the unit vector sin(68.04)j+ cos(68.04)k. Now, you can find the angle between those vectors by using the dot product and the fact that $u\cdot v= |u||v|cos(\theta)$ where $\theta$ is the angle you want to find.