# Plane Vectors

About a year ago, I was trying to teach my students about vector addition when I found this video on CNN, which does a wonderful job of depicting vector addition problems in real life. The plane is trying to fly to its right, but yet it is actually moving straight down the runway.

The impetus for the straight-ahead movement, we are to understand, is a leftward-blowing wind, which balances out the rightward-pointing plane.

While we don’t know exactly how strong the wind is, we can estimate it fairly easily. My students estimated that the plane was pointing $30^{\circ}$ to the right. A simple internet search told us that large planes land at speeds between $120$ and $150$ miles per hour. Right triangle trigonometry tells us that the wind speed, then, is between $\approx 120\sin(30^{\circ})$ and $\approx 150\sin(30^{\circ})$, or between roughly $60$ and $75$, miles per hour. Winds of $75$ miles per hour are characteristic of weak (F1) tornadoes, but $60$ mile per hour winds are regular enough to be believable here. (Data from Weather Underground confirm that maximum wind speeds were in the neighborhood of $55$ to $65$ miles per hour in the Chicago area in the days before that video was posted.)

However, this does not tell the complete story about the wind. We are only able to “see” the portion of the wind blowing across the runway. The wind might not be going directly perpendicularly to the runway, though. It might be blowing a little bit into the plane, slowing it down, or it may be a bit of a tailwind, speeding the plane up. What we can say is the the component of the wind moving perpendicular to the runway is the $60$ to $75$ miles per hour we had found.

So, if we remember our formulas that $$\mathrm{comp}_{\vec{w}}(\vec{v})=\frac{\vec{v}\cdot\vec{w}}{|\vec{w}|},$$ we can improve our estimates on the wind speed.