Introduction to College Math · Project 1 · Option 2
Vector Bench
A solar installer has to know which way a panel faces, and how much of the sunlight is actually perpendicular to the glass. Coordinates, a cross product and a dot product answer both questions for one simplified roof.
Why vectors
A single tilt angle cannot tell south from east. Two edge vectors fix the rectangle in space. Their cross product is the direction the glass faces, and the dot product with the sun says whether that face is looking at the light.
Operations in the model
- Subtraction. B − A and D − A are the two edges lying in the glass.
- Dot product. u · v checks the right angle. n · s measures how directly the sun meets the panel.
- Cross product. u × v is the outward normal, and its length is the area.
- Scalar multiplication. Dividing the normal by its length gives a unit direction for the angle.
- Addition. The sun vector is the sum of its east, north and upward components.
Case study
Eaves of 3 m run east. The other edge runs 4 m north and rises 3 m, so it is 5 m long and the panel is a 3 by 5 rectangle. The sun points south and up, s = ⟨0, −4, 3⟩.
- n = u × v
- ⟨0, −9, 12⟩
- Area
- 15 m²
- cos θ
- 24/25
- θ
- 16.26°