I have the answer to this problem but can not figure out the math or
rationale to make this work:

Dan drives 1 mile north. then .50 mile east, 1.0 mile south, 2.0 miles
west, then .50 miles northwest and 0.75 miles northeast. What is the
magnitude and direction of  Dan's displacement vector at the end of
the trip?

The answer is D vector =1.6 miles and direction is 57 degrees West of
North.

Please show vector diagram and all calculations used.

Hi,

We can attack the question by first calculating the X and Y components
of the movement. I am assuming the directions in the usual way, ie
north is positive y, and east is positive x direction.

First, please note that the length of X (or Y, for that purpose)
component of a 45 degree vector (NW, NE etc) is:

|X| = |Y| = length * cos(45)


Let's start with adding up the X and Y components:

X = 0.5 - 2 - 0.5 * cos(45) + 0.75 * cos(45) = -1.5 + 0.25 * cos(45) =
-1.323223

Y = 1 - 1 + 0.5 * cos(45) + 0.75 * cos(45) = 1.25 * cos(45) =
+0.883883


So, Dan effectively drove 0.883883 miles to north and 1.323223 miles
to west.


Total displacement can be calculated using Pythagoras' equation:

d^2 = (1.323223)^2 + (0.883883)^2
d = 1.6 miles


The angle alpha can be calculated using:

tan(alpha) = 1.323223 / 0.883883

alpha = 56.25 degrees


Note: You can find a diagram summarizing the vectors at:

http://www.geocities.com/bio_ga/156453.html


Hope this helps
Regards
Bio

---

Request for Answer Clarification by plato1000-ga on 18 Feb 2003 14:34 PST

Can you restate the solution using geometry and trig vs. using components?

---

Clarification of Answer by bio-ga on 18 Feb 2003 18:40 PST

Please refer to the same URL as the previous answer (reload the page
if necessary), and look at the bottom for the words "DIAGRAM FOR
ALTERNATE ANSWER":

If you compare the diagram with the previous answer, it is obvious
that:
A = 1.5
B = 0.5
D = 0.75
c = 90 + 45 = 135 deg

Using cosine theorem:

C^2 = A^2 + B^2 - 2AB*cos(c)
C = 1.887

and using sine theorem:

(sin a) / A = (sin b) / B = (sin c) / C

as we know c, C, A and B, we can easily calculate a and b angles:
a = 34.2 deg
b = 10.8 deg

From the diagram:
a + e = 90 deg
e = 90 - a = 55.8 deg

Using the cosine theorem again to calculate E:

E^2 = C^2 + D^2 - 2CD*cos(e)

E = 1.587 miles ***

Using the sine theorem:

(sin e) / E = (sin d) / D
d = 23.0 deg

alpha = 90 - (b + d)
alpha = 56.2 deg

Hope this helps
Regards
Bio