I think a good way to think of this intuitively is to see cos(theta) * A as the difference between the the right most and left most points on the surface. This may also simplify some calculations if that would be easier to compute.
jgforsberg
Intuitively we are projecting the surface into the light beam to see how much of the surface is hit by the light. Recall we can get the proportion of the surface in the light by projecting the surface into the configuration of the first cube. This scalar projection is done by taking a unit vector parallel to the surface and doing the inner product with a unit vector that is perpendicular to l and thus in the entire light beam (like the first cube). Since the dot product of two vectors equals the dot product of the two vectors rotated 90 degrees counter-clockwise, we can say the projection is proportional to the dot project between l and n. Recall the dot product is proportional to cosine of the angle between two vectors. This gives us a mathematical justification for why the irradiance is proportional to the cosine theta.
jgforsberg
Intuitively we are projecting the surface into the light beam to see how much of the surface is hit by the light. Recall we can get the proportion of the surface in the light by projecting the surface into the configuration of the first cube. This scalar projection is done by taking a unit vector parallel to the surface and doing the inner product with a unit vector that is perpendicular to l and thus in the entire light beam (like the first cube). Since the dot product of two vectors equals the dot product of the two vectors rotated 90 degrees counter-clockwise, we can say the projection is proportional to the dot project between l and n. Recall the dot product is proportional to cosine of the angle between two vectors. This gives us a mathematical justification for why the irradiance is proportional to the cosine theta.
I think a good way to think of this intuitively is to see cos(theta) * A as the difference between the the right most and left most points on the surface. This may also simplify some calculations if that would be easier to compute.
Intuitively we are projecting the surface into the light beam to see how much of the surface is hit by the light. Recall we can get the proportion of the surface in the light by projecting the surface into the configuration of the first cube. This scalar projection is done by taking a unit vector parallel to the surface and doing the inner product with a unit vector that is perpendicular to l and thus in the entire light beam (like the first cube). Since the dot product of two vectors equals the dot product of the two vectors rotated 90 degrees counter-clockwise, we can say the projection is proportional to the dot project between l and n. Recall the dot product is proportional to cosine of the angle between two vectors. This gives us a mathematical justification for why the irradiance is proportional to the cosine theta.
Intuitively we are projecting the surface into the light beam to see how much of the surface is hit by the light. Recall we can get the proportion of the surface in the light by projecting the surface into the configuration of the first cube. This scalar projection is done by taking a unit vector parallel to the surface and doing the inner product with a unit vector that is perpendicular to l and thus in the entire light beam (like the first cube). Since the dot product of two vectors equals the dot product of the two vectors rotated 90 degrees counter-clockwise, we can say the projection is proportional to the dot project between l and n. Recall the dot product is proportional to cosine of the angle between two vectors. This gives us a mathematical justification for why the irradiance is proportional to the cosine theta.