Vector Problem

**jessicadowning** · September 30th 2012, 10:04 AM

An airplane flies horizontally from east to west at 304 mi/hr relative to the air. If it flies in a steady 24 mi/hr wind that blows horizontally toward the southwest (45 degrees south of west), find the speed and direction of the airplane relative to the ground.

**Soroban** · September 30th 2012, 11:29 AM

Hello, Jessica!

An airplane flies horizontally from east to west at 304 mi/hr relative to the air.
If it flies in a steady 24 mi/hr wind that blows horizontally toward the southwest (45^o south of west),
find the speed and direction of the airplane relative to the ground.

Code:

                         304
        - - - B *   *   *   *   *   * A
          45^o * 135^o          *
            *           *
       24 *       *
        *   *
     C*

The plane is flying from $A$ to $B$ at 304 mph.
But there is wind blowing from $B$ to $C$ at 24 mph.
Hence, the plane's flight is from $A$ to $C.$

Law of Cosines:
. $AC^2 \;=\;24^2 + 304^2 - 2(24)(304)\cos135^o \;=\; 103,\!210.1002$

Hence: . $AC \;=\;321.4188889 \;\approx\;321.4\text{ mph.}$

Law of Cosines:
. . $\cos A \;=\;\frac{304^2 + 321.4^2 - 24^2}{2(204)(321.4)} \;=\;0.998601718$

Hence: . $A \;=\;3.030300248 \;\approx\;3^o$

The direction is 3^o south of west.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Um . . . it said vectors, didn't it?

$\overrightarrow{AB} \;=\;\langle 304\cos180^o,\:304\sin180^o\rangle \;=\;\big\langle -304,\:0\big\rangle$

$\overrightarrow{BC} \;=\;\langle 24\cos225^o,\:24\sin225^o\rangle \;=\;\left\langle \text{-}12\sqrt{2},\:\text{-}12\sqrt{2}\right\rangle$

$\overrightarrow{AC} \;=\;\overrightarrow{AB} + \overrightarrow{BC} \;=\; \big\langle \text{-}304-12\sqrt{2},\:\text{-}12\sqrt{2}\big\rangle$

Then: . $|\overrightarrow{AC}| \;=\;\sqrt{\left(\text{-}304 - 12\sqrt{2}\right)^2 + \left(\text{-}12\sqrt{2}\right)^2}$

Hence: . $|\overrightarrow{AC}| \;=\;\sqrt{103,\!310.1022} \;=\;321.418889 \;\approx\;321.4\text{ mph.}$

$\tan\theta \;=\;\frac{\text{-}12\sqrt{2}}{\text{-}304-12\sqrt{2}} \;=\;0.052872695$

. . $\theta \;=\;3.026561265 \;\approx\;3^o$

I'm gratified that my answers match!

**jessicadowning** · September 30th 2012, 11:46 AM

Thanks, this really helped!

Thread: Vector Problem

LinkBack

Thread Tools

Display

Vector Problem

Re: Vector Problem

Re: Vector Problem

Similar Math Help Forum Discussions

Vector problem 2

Vector Problem

[SOLVED] Vector problem

Vector problem

vector problem

Search Tags