How Do You Calculate Displacement from Dallas to Chicago Using Vector Addition?

In summary, the homework statement is that Atlanta is 730 miles north of east from Dallas, and Chicago is 559 miles north of east from Atlanta. The same map also shows that Chicago is 21.0° west of north from Atlanta. To find the displacement from Dallas to Chicago, you used the trig functions and the Pythagorean theorem. The displacement is 719.542 miles, and the angle is 41.9 degrees northeast of Dallas.
  • #1
bdh2991
103
0

Homework Statement



A map suggests that Atlanta is d1 = 730 mi in a direction of θ1 = 5.12° north of east from Dallas. The same map shows that Chicago is d2 = 559 miles in a direction of θ2 = 21.0° west of north from Atlanta. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago.

Homework Equations



trig functions

The Attempt at a Solution



i used the vector <65.1466, 727.087> for dallas to atlanta, and used the vector <521.87, 200.328> for atlanta to chicago. I added them together to get vector from dallas to chicago, then used the pythagorean theorem to get the resultant magnitude which was 719.542. The angle i got was 32.33 degrees by using inverse tangent of the dallas to chicago vector, then subtracted the 5.12 degrees to get the angle northeast of dallas. Still saying i have the wrong answer so I'm not sure what I'm doing wrong. please help
 
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  • #2
bdh2991 said:
i used the vector <65.1466, 727.087> for dallas to atlanta, and used the vector <521.87, 200.328> for atlanta to chicago.
To start with, it looks like you have mixed up your x and y components. (Assuming you mean it as x, y, where +x is east and +y is north.) And you need to be careful with signs.
 
  • #3
i see, so correcting my mistakes i should use <-200.3277, 521.8715> as my second vector. working the rest of the problem out accordingly gives me the resultant 788.7116 and the angle 41.9 degrees. i hope this is right, and also I'm still confused if i should use 41.9 degrees as the angle they are asking for or 41.9-5.12 = 36.78 as my answer?
 
  • #4
bdh2991 said:
i see, so correcting my mistakes i should use <-200.3277, 521.8715> as my second vector. working the rest of the problem out accordingly gives me the resultant 788.7116 and the angle 41.9 degrees.
What's that angle measured from?
i hope this is right, and also I'm still confused if i should use 41.9 degrees as the angle they are asking for or 41.9-5.12 = 36.78 as my answer?
Why would you subtract 5.12 ?

Express the angle as X degrees north of east (or equivalently, Y degrees east of north).
 
  • #5
the angle would be 48.1 degrees north of east, correct?
 
  • #6
bdh2991 said:
the angle would be 48.1 degrees north of east, correct?
That sounds about right.
 

Related to How Do You Calculate Displacement from Dallas to Chicago Using Vector Addition?

1. What is a vector resultant problem?

A vector resultant problem involves finding the resultant of two or more vectors, taking into account both magnitude and direction. It is commonly used in physics and engineering to solve problems involving forces, velocities, and displacements.

2. How do you find the resultant of two vectors?

To find the resultant of two vectors, you can use the parallelogram method or the head-to-tail method. In the parallelogram method, you draw the two vectors as adjacent sides of a parallelogram and the resultant is the diagonal of the parallelogram. In the head-to-tail method, you draw the first vector and then the second vector starting from the end of the first vector. The resultant is the vector that starts from the beginning of the first vector and ends at the end of the second vector.

3. Can the resultant of two vectors ever be smaller than the individual vectors?

No, the resultant of two vectors can never be smaller than the individual vectors. It will always have a magnitude that is equal to or greater than the magnitude of the individual vectors.

4. How do you calculate the magnitude of the resultant vector?

The magnitude of the resultant vector can be calculated using the Pythagorean theorem, which states that the square of the hypotenuse (the resultant vector) is equal to the sum of the squares of the other two sides (the individual vectors). In other words, the magnitude of the resultant vector is the square root of the sum of the squares of the magnitudes of the individual vectors.

5. What is the difference between the resultant and the equilibrant of a system of vectors?

The resultant is the vector that represents the net effect of all the individual vectors in a system, while the equilibrant is the vector that exactly cancels out the effect of the resultant. In other words, the equilibrant is equal in magnitude and opposite in direction to the resultant, resulting in a net force or displacement of zero.

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