Corners of the glass
x east, y north, z up, in metres
A = ⟨0, 0, 0⟩
B = ⟨3, 0, 0⟩
D = ⟨0, 4, 3⟩
C = B + v = ⟨3, 4, 3⟩
These four points are the corners of one rectangular panel. A is the local origin, not a point on a map.
Edges, by subtraction
u = B − A = ⟨3, 0, 0⟩ − ⟨0, 0, 0⟩ = ⟨3, 0, 0⟩
v = D − A = ⟨0, 4, 3⟩ − ⟨0, 0, 0⟩ = ⟨0, 4, 3⟩
|u| = 3 m
|v| = 5 m
u lies along the eaves. v runs from the eaves up the slope. Subtraction turns two corner positions into a direction in the glass.
Right angle, by a dot product
u · v = 3·0 + 0·4 + 0·3 = 0
The dot product is zero, so the edges are perpendicular and the panel is a rectangle. That happens because v has no east component: the eaves stay east–west.
Normal, by the cross product
n = u × v
x = 0·3 − 0·4 = 0
y = −( 3·3 − 0·0 ) = −9
z = 3·4 − 0·0 = 12
n = ⟨0, −9, 12⟩
u · n = 0
v · n = 0
n is perpendicular to the glass. The sign of the north component says which way the front faces: negative looks south, positive looks north. The upward component says the front still sees the sky.
Area, from the length of the normal
|n| = |east| √(north² + rise²) = 3 √(4² + 3²) = 3 √25 = 15
Area = |u × v| = 15 m²
For any parallelogram the area is the length of the cross product. Here the right angle makes that length equal to |u| times |v|.
Unit normal, by scalar multiplication
n̂ = (1/15) ⟨0, −9, 12⟩ = ⟨0, −3/5, 4/5⟩
Dividing by the length keeps the direction and throws away the area. The angle with the sun should not change if the panel is measured in centimetres instead of metres.
Sun direction, by addition
s = ⟨0, 0, 0⟩ + ⟨0, −4, 0⟩ + ⟨0, 0, 3⟩
s = ⟨0, −4, 3⟩
|s| = 5
s points from the panel toward the sun. It is the sum of an east piece, a north piece and an upward piece. Only the direction matters, so a longer arrow is the same sun.
Alignment, by the dot product
n · s = 0·0 + (−9)·(−4) + 12·3 = 72
cos θ = (n · s) / (|n| |s|) = 72 / (15 × 5) = 72/75 = 24/25
θ = arccos(cos θ) ≈ 16.26°
Front-face share = max(cos θ, 0) = 24/25
cos θ is the share of a direct beam that lands perpendicular to the glass. Positive means the sun is on the front. Zero means the beam skims the surface. Negative means the sun is behind the panel, and the front-face share is taken to be zero.
Compare with a flat roof
k = ⟨0, 0, 1⟩
k · ŝ = 3/5
(24/25) / (3/5) = 8/5 = 1.60
A flat roof has normal k = ⟨0, 0, 1⟩. The ratio says how many times more of this same direct beam the tilted panel takes, compared with lying flat.
Check against the south-facing tilt rule
α = β = arctan(3/4), and sin(α + β) = sin(2α) = 2 × (3/5) × (4/5) = 24/25
When the sun is due south and the panel faces south, PVEducation’s sin(α + β) is the same number as this dot product. α is the sun’s elevation and β is the tilt from horizontal.