Some really confusing directions (and the world is flat).

[niyati]

A map suggests that Atlanta is 730 miles in a direction of 5.00 degrees north of east from Dallas. The same map shows that Chicago is 560 miles in a direction of 21.0 degrees west of north from Atlanta. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.

Equations:

Ax = Acos(angle)
Ay = Asin(angle)

(same for vector B)

[resultant vector] = ((Ax + Bx)^2 + (Ay + By)^2)

So far, I graphed each pathway on the same picture, so that the resultant vector would complete a triangle. From Dallas to Atlanta (vector A), the h.c. = 730cos85, and the v.c. = 730sin85.

For Atlanta to Chicago (vector B), I couldn't figure out which angle to use. 69 degrees or 111? I'm thinking 111 degrees, because if I draw the three cities, so that two tails of vectors stem from Atlanta and form a V-shape (as if the positive y-axis is equivalent to north, the positive x-axis is equivalent to east on a compass, and negative x-axis to west), the angle would be 111 degrees.

If this is true, I was planning on getting the h.c. and v.c of B, and adding up the h.c.'s of A and B, and the v.c.'s of A and B, to the the total h.c. and v.c of R. Once that is given, I can use Pythagorean Theorum to get the magnitude of R, which is the total displacement between Dallas and Chicago.

 

[PhanthomJay]

A map suggests that Atlanta is 730 miles in a direction of 5.00 degrees north of east from Dallas. The same map shows that Chicago is 560 miles in a direction of 21.0 degrees west of north from Atlanta. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.

Equations:

Ax = Acos(angle)
Ay = Asin(angle)

(same for vector B)

[resultant vector] = ((Ax + Bx)^2 + (Ay + By)^2)

So far, I graphed each pathway on the same picture, so that the resultant vector would complete a triangle. From Dallas to Atlanta (vector A), the h.c. = 730cos85, and the v.c. = 730sin85.

For Atlanta to Chicago (vector B), I couldn't figure out which angle to use. 69 degrees or 111? I'm thinking 111 degrees, because if I draw the three cities, so that two tails of vectors stem from Atlanta and form a V-shape (as if the positive y-axis is equivalent to north, the positive x-axis is equivalent to east on a compass, and negative x-axis to west), the angle would be 111 degrees.

If this is true, I was planning on getting the h.c. and v.c of B, and adding up the h.c.'s of A and B, and the v.c.'s of A and B, to the the total h.c. and v.c of R. Once that is given, I can use Pythagorean Theorum to get the magnitude of R, which is the total displacement between Dallas and Chicago.

You are using 85 degrees as the angle of vector A, which is the angle the vector makes with the vertical; the angle is 5 degrees above the horizontal (it is given that the vector is 5 degrees north of east). If you choose the x component as the cosine of the angle, you best use 5 degrees in your calcs. For vector B, you can use 111 degrees, and the plus or minus sign for the components will come out of that equation; or if you use 69 degrees, that's OK also, but then you have to decide using a sketch as to whether the components are plus or minus.

 

[niyati]

Oh crap. I think I drew my picture wrong. The way I drew it was as if the problem said, "5.00 degrees east of north", which is, well, WRONG.

I'll redraw it and see what I come up with.

Thank you!

 

[niyati]

Uhm, okay.

So, Rx = [730cos(5 degrees) + 560cos(111 degrees)], and Ry = [730sin(5 degrees) + 560sin(111 degrees)].

Thus, the resultant vector's magnitude would be the square root of both terms squared (which I actually forgot to add to the equation I put into my post).

Would this be correct?

 

[PhanthomJay]

Yes, this would give you the correct magnitude of the resultant displacement. But you must now calculate the direction of the resultant displacement.