### Video Transcript

Part of a concave lens is shown. No light rays pass through the lens, but several possible lines normal to the surface of the lens are shown. Which line on the left-hand side of the lens is normal to the surface at the point that the line contacts the surface? Which line on the right-hand side of the lens is normal to the surface at the point that the line contacts the surface?

Alright, let’s take a look at our diagram which, we’re told, shows us part of a concave lens. On either side of the lens, we can see four dashed lines drawn in. On the left, lines A, B, C, and D and on the right, lines E, F, G, and H. Looking first at the left-hand side of the lens, we want to figure out which of these four lines — A, B, C, and D — is normal to the surface of the lens at the point where the lines cross that surface. And then, the same thing for the right-hand side, which of these four lines — E, F, G, and H — is normal to the lens surface at the point where the lines cross that surface?

As we get started, let’s clear a bit of space on screen and remind ourselves what it means for lines to be normal to one another. When two lines are perpendicular — so there’s a 90-degree angle between them — we say that they’re normal to one another. Now, we want to find which of these four lines is normal to the surface of this lens, which at first may seem confusing because we see the lens surface is curved. But it’s still possible for a line to be normal to a curved surface. We would just say that it’s normal to that surface at a particular point. The point we’re interested in is this one right here, where the four lines intersect the surface.

To figure out which of lines A, B, C, and D is normal to the surface of the lens at this point, we’ll draw what’s called a tangent line to the surface that crosses through that point. This line we’ve drawn in indicates the slope or the gradient of the lens surface at the point of interest. So whichever of lines A, B, C, and D is normal to the tangent is also normal to the surface of the lens at the point where the line contacts the surface. As we consider these four candidates, once again, we’re looking for lines that cross at 90 degrees. We can see that the angle between the tangent and line A is less than 90 degrees. Those, therefore, can’t be normal.

The same is true for line B and even for line C, which looks to be a horizontal line, which would mean that, for it to be normal to our tangent line, then our tangent would need to be vertical. But we can see that it’s not. This leaves us with line D. If we trace in line D, to get a better sense of how it crosses the tangent, we see that, indeed, it looks to be at 90 degrees to that tangent. In other words, it’s normal to it, which means it’s normal to the surface of the lens at the point where that line contacts the surface. So our answer to the first question is line D. This is the line normal to the left-hand surface of the lens at the point where the line in the surface meet.

Now we want to answer the same question, but about the right-hand side of the lens. Once again, we’re considering the point where these four candidate lines cross the lens surface. And just like before, we’ll draw a line tangent to the surface at this point. We’re looking for which of the lines — E, F, G, and H — is at 90 degrees or perpendicular to this tangent we’ve drawn in. Considering the angle between the tangent line and line E, we can see that line E won’t be it. Neither will line F. The angle between line F and the tangent is clearly less than 90 degrees. And starting on the other side of the tangent line, if we go from there to line H, that angle looks to be less than 90 as well.

This leaves line G. Once again, we’ll trace in this line to clearly see where it crosses the tangent. We can now see that line G is indeed at 90 degrees to the tangent line we’ve drawn, which means it’s normal to that line, which means it’s normal to the surface at the point that the line contacts the surface. This then is our answer to the second part of the question. The line on the right-hand side of the lens, normal to the surface, at the point that the line contacts the surface is line G.