# a man walks 7.0miles at 55 degrees north of west. then turns and walks 8.5miles at east. he turns again and walks 2.0 miles at 64 degrees north of east. what is his displacement

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Displacement is the distance from the end point and the starting point.

To calculate that, we'll find the distances travel in the x and y direction during all three directions of movement.

Then we'll combine movements in the x direction and combine movements in the y direction.

After that we'll use the distance formula to find the distance between the origin (starting point) and the end point.

See images :)

Note: Picture not drawn to scale

Another note: 55 degrees North of West means start in west direction and move 55 degrees to the north. 64 degrees North of east means start at east direction and move 64 degrees North.

**Images:**

One way to imagine this question is as a quadrilateral. You're given 3 sides and you've been asked to calculate the remaining side.To do this, sum up the 3 sides you know, and what remains will be the remaining side.

I like at times like this, it's often easier to split up the displacement north from the displacement east as each line can be considered as the hypotenuse of a right-angled triangle with the other two sides pointing north and east. This requires trigonometry to work out.

Remember the rule for the trig functions SohCahToa. `sin(angle)=((opposite)/(hipotenuse))`

`cos(angle)=((adjacent)/(hypotenuse))`

`tan(angle)=((opposite)/(adjacent))`

Take the first line (which I'm going to enotre with a subscript 1), 7 miles at 55° North of West. To work out how far along the North axis that will take him, see that we have the angle opposite to the side we want to know about and the hypotenuse. We therefore need the Sine trig function.

Plugging in the details above, we get

`sin(55^@)=((?_1_n)/(7))`

which can then be rearranged to

`?_1_n=7*sin(55^@)` miles north

Doing the same with the West component using the Cos function gives:

`?_1__e=-7*cos(55^@)`

As I am using East for the coordinate direction, any distance travelled West is considered negative.

For the second line (8.5m east), there is no north component, only east. Therefore, the North component is 0 and the East component is 8.5m.

For the third line, (2m at 64° north of east), the same applies. Thus the North component is

`?_3_n=2*sin(64^@)`

and the East component is

`?_3_e=2*cos(64^@)`

At this point we have all the information we need. To get the North component of the displacement, sum the 3 North components together.

`?_n=7*sin(55^@)+0+2*sin(64^@)`

`?_n~~7.53` miles

and sum the 3 East components together to get the east component of the displacement.

`?_e=-7*cos(55^@)+8.5+2*cos(64^@)`

`?_e~~5.36` miles

**Sources:**