Let's see how vector calculus can be used to give a short, easy proof of the important reflective property of the parabola, namely, that if a ray of light travels parallel to the symmetry axis of a parabola (or in three dimensions, a paraboloid of revolution) and strikes the concave side of the parabola, then it will be reflected to the focus. Equivalently, if a ray of light leaves the focus and strikes the parabola, it will be reflected in a path parallel to the symmetry axis. This property is an essential feature of satellite dishes, car headlights, radio telescopes, and reflecting telescopes, including liquid mirror telescopes.
Before giving the proof, let's notice that if w = w(t) is a vector function of time t, then we can use the definition of vector magnitude and the dot product rule to compute
Now let's assume that we have a parabola with its focus at the origin 0 and a horizontal symmetry axis, and that the parabola is given by a smooth parameterization r = r(t), where r is the position vector of a point on the curve and t is time (see figure below).
If we think of a light ray leaving the focus, traveling along r to the red dot on the curve, and striking the curve at an angle of incidence 
to the tangent line at that point, then we want to show that when it bounces off the curve, it travels horizontally (in the direction of the horizontal unit vector u). By the law of reflection (angle of incidence = angle of reflection) this means that we must show that 
= . Since and are between 0 and 180 degrees, this is equivalent to showing that cos = cos.
Now here's the proof, which uses the notation in the figure.
By the definition of a parabola, we know that
or equivalently,
u = ||r||Where L = (distance from directrix to vertex) = (distance from vertex to focus). Next, we take derivatives of both sides of this equation, keeping in mind that 2L is a constant and u is a constant vector (their time derivatives are zero). Also, we'll need Equation (*) when taking the derivative of ||r||. We get
and since u and r / ||r|| are unit vectors, this implies that cos = cos.
This elegant little proof is due to Professor Harley Flanders. It appears in the article
H. Flanders, The optical property of the conics, Amer. Math. Monthly, 75 (1968), 399.
This issue of the American Mathematical Monthly is available in the IUPUI University Library.
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