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# Solution: You leave the airport in College Station and fly 24.0 km in a direction 34.0° south of east. You then fly 46.0 km  m kmdue north.How far must you then fly to reach a private landing strip that is 32.0

###### Problem

You leave the airport in College Station and fly 24.0 km in a direction 34.0° south of east. You then fly 46.0 km due north.

How far must you then fly to reach a private landing strip that is 32.0 km due west of the College Station airport?

In what direction?

###### Solution

Steps for finding a missing vector:

1. Write the vector equation for the problem (drawing a diagram usually helps a lot!)
2. Decompose the given vectors and organize known information in a table.
3. Write algebraic equations and solve.
4. Find magnitude and angle of the resultant if required.

When writing a vector equation for one of these problems, it'll be in a form like

$\overline{)\stackrel{⇀}{A}{+}\stackrel{⇀}{B}{+}{.}{.}{.}{=}\stackrel{⇀}{R}}$,

where the missing vector is one of the vectors on the left-hand side of the equation. Note that if the problem is about returning to the starting point, $\stackrel{⇀}{R}=\stackrel{⇀}{0}$.

For decomposing given vectors, we'll have to define a coordinate system if the problem hasn't, then use the equations

If the final answer is supposed to be in magnitude-angle notation, we'll need to use these equations to get the magnitude and direction:

$\overline{)|\stackrel{⇀}{A}|{=}\sqrt{{{A}_{x}}^{2}+{{A}_{y}}^{2}}}$

Step 1. This problem has the plane flying three separate legs (24.0 km at 34.0° south of east, 46.0 km north, and the unknown). We'll call them $\stackrel{⇀}{A}$, $\stackrel{⇀}{B}$, and $\stackrel{⇀}{C}$ in order. The resultant $\stackrel{⇀}{R}$ is 32.0 km west. So our vector equation is

$\overline{)\stackrel{⇀}{A}{+}\stackrel{⇀}{B}{+}\stackrel{⇀}{C}{=}\stackrel{⇀}{R}}$

Step 2. Define a coordinate system so that north is the +y-direction and east is the +x-direction. Then we can decompose vector $\stackrel{⇀}{A}$ into x- and y-components. "South of east" in this coordinate system is counterclockwise from the +x-axis, so

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