What is the magnitude of the tension on the ends of the clothesline?

[undrcvrbro]

Homework Statement

A horizontal clothesline is tied between 2 poles, 12 meters apart.
When a mass of 2 kilograms is tied to the middle of the clothesline, it sags a distance of 5 meters.

What is the magnitude of the tension on the ends of the clothesline?

Homework Equations

F=mg
g=9.8 m/s^2

The Attempt at a Solution

Because the two ends of the rope have the same vertical direction but opposite horizontal direction:

T1 = (-a)i + (b)j
T2 = (a)i + (b)j

Using similar triangles:

b/a = (5m) / (6m) => a = (6/5)b

Solving for the weight of the 2 kg mass:

F = (2 kg) * (9.8 m/s^2) = 19.6 N

T1 + T2 = -w

( (-a)i + (b)j ) + ( (a)i + (b)j ) = 19.6 j

substituting (6/5)b for a and simplifying:

( (-6/5b)i + (b)j ) + ( (6/5b)i + (b)j ) = 19.6 j

(2b)j = 19.6j
b = 9.8

a = (6/5)b = (6/5)*(9.8) = 11.76

T1 = (-a)i + (b)j = (-11.76)i + (9.8)j
T2 = (a)i + (b)j = (11.76)I + (9.8)j

So I have that much. But my only issue is that this is an online problem and it only has one spot for an entry. From what I can tell, he wants the final answer to be a numerical one in Newtons.

Did I do a whole bunch of unnecessary math, or is my answer hidden somewhere in my work(i.e. I'm just not completely understanding the conceptual side of the math I'm doing)?

Besides, as I said, there is only one entry area. So I'm assuming he only wants one answer. So that would mean the Tension on each end of the line is equal. Is that true in this case?

 

[nicksauce]

It asks for the magnitude (ie. the length of the vector) of the tension at the ends of the clothesline. The magnitude of (-11.76)i + (9.8)j is certainly equal to the magnitude of (11.76)i + (9.8)j.

 

[undrcvrbro]

It asks for the magnitude (ie. the length of the vector) of the tension at the ends of the clothesline. The magnitude of (-11.76)i + (9.8)j is certainly equal to the magnitude of (11.76)i + (9.8)j.

First of all, thanks for responding!

Wait, so should I just calculate the magnitude of either vector? So for instance, if I were to use T1...

abs(T1) = $$\sqrt{(-11.76^2) + (9.8^2)}$$

So T1= 15.3

Edit: I just submitted my answer using that method and it's correct. Thanks again.