# Thread: Vector question (relative speed)

1. ## Vector question (relative speed)

A plane goes from city A to city B. In a Cartesian plane, city A is at the origin and city B has coordinates (100,150). If there is no wind, the flight lasts one hour. Unfortunately, there is a wind. If the pilot does not adjust his flight path, he will be at point (120,160) after an hour. What is the speed of the wind?

I am having trouble doing this question, and my teacher has warned there will be a question similar to this on our test. I would like to know the steps I need to take in order to solve this question.

2. Hello, Solid8Snake!

A plane goes from city A to city B. In a Cartesian plane.
City A is at the origin and city B has coordinates (100,150).
If there is no wind, the flight lasts one hour. Unfortunately, there is a wind.
If the pilot does not adjust his flight path, he will be at point (120,160) after an hour.
What is the speed of the wind?
Code:
      |                             C
|                             o(120,160)
|                          * *
|                       *   *
|                    *     *
|                 *       *
|              *         *
|           *           o(100,150)
|        *        *     B
|     *     *
|  *  *
A * - - - - - - - - - - - - - - - - - - -
|

The plane's original vector was: .$\displaystyle \overrightarrow{AB} \:=\:\langle100,150\rangle$

Due to the wind, $\displaystyle \overrightarrow{BC} \:=\:\langle x,y\rangle$, it flies to $\displaystyle C(120,160).$

Since $\displaystyle \overrightarrow{AB} + \overrightarrow{BC} \:=\:\overrightarrow{AC}$, we have: .$\displaystyle \langle 100,150\rangle + \langle x,y\rangle \:=\:\langle120,160\rangle$

. . Hence: .$\displaystyle \begin{array}{ccccccc}100 + x \:=\:120 & \Longrightarrow & x \:=\:20 \\ 150 + y \:=\:160 & \Longrightarrow & y \:=\:10 \end{array}$

The wind's vector is: .$\displaystyle \overrightarrow{BC} \:=\:\langle 20,10\rangle$

Its speed (magnitude) is: .$\displaystyle |\overrightarrow{BC}| \;=\;\sqrt{20^2+10^2} \;=\;\sqrt{500} \;=\;10\sqrt{5}$

3. thx a million