Animated Solution for Physics - Optics: The incident ray, reflected ray and the outward drawn normal are denoted by the unit vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$, respectively. Then, choose the correct relation for these vectors.

Have you ever wondered how to express the simple law of reflection—"angle of incidence equals angle of reflection"—using the elegant language of vectors? This problem asks us to find the exact mathematical relationship between the incident ray $\mathbf{a}$, the reflected ray $\mathbf{b}$, and the outward drawn normal $\mathbf{c}$.

Let's dive into the geometry and see how beautifully vectors handle this physical phenomenon.

To make our analysis straightforward, let's set up a 2D Cartesian coordinate system. Imagine the mirror lying flat along the x-axis. The outward drawn normal vector $\mathbf{c}$ points straight up. Since we are told $\mathbf{c}$ is a unit vector, we can simply write it as:

Now, let's look at the incident ray $\mathbf{a}$. It strikes the mirror at an angle of incidence $\theta$ with respect to the normal. Because it is traveling towards the mirror, it moves forward along the x-axis but downward along the y-axis. Breaking it into components, we get:

Similarly, the reflected ray $\mathbf{b}$ bounces off at the exact same angle $\theta$. It continues its forward journey along the x-axis, but its vertical motion is now directed upward, away from the mirror:

Our goal is to find an operation that transforms vector $\mathbf{a}$ into vector $\mathbf{b}$. Notice that the horizontal component ($\sin \theta \hat{i}$) remains completely unchanged, while the vertical component ($-\cos \theta \hat{j}$) perfectly flips its sign to become ($+\cos \theta \hat{j}$).

Let's test the relation given in option (c): $\mathbf{b}=\mathbf{a}-2(\mathbf{a}\cdot \mathbf{c})\mathbf{c}$.

First, we calculate the dot product of the incident ray and the normal. This operation isolates the vertical component of the incident ray:

Now, we substitute this back into our master equation:

The two negative signs cancel out, giving us a positive term:

And there it is! This perfectly matches our expression for the reflected ray $\mathbf{b}$.

Conclusion: This vector equation is the universal, coordinate-free form of the law of reflection. It elegantly states that to find a reflected ray, you take the incident ray and reverse its component that is perpendicular to the mirror, leaving the parallel component untouched.