Find the tension on the strings using vectors.
This is such an easy problem for me when I was in my physics class, we just summed up the forces in the x-direction set it to the force of the weight in the x, and did the same for y.
When introduced into vectors in calc 3 I realized hey it's basically the same thing. However the notion of a normalize vector really threw me off. Apparently a vector isn't correct if its magnitude isn't 1, if that's the case you have to divide it by it's magnitude
-R = T1 + T2
The Attempt at a Solution
I found the x components of the triangles using the pythagorean theorem.
√x2+(3/4)2 = 2
x = ± 1/4 √55
√x2+(3/4)2 = 1
x = ± 1/4 √7
I took the negative answer of 1/4 √7, because I am assuming that right is positive so the x-component is left, thus negative.
-200<0,-1> = T1 * 2<1/4√55, 3/4> + T2 * 1<-1/4√7, 3/4>
At this point I am not sure, it's my understanding from physics that you don't distribute the magnitudes into the vectors, only if they are in terms of cosine and sin. So I decided that magnitude of 2 was somehow already incorporated into 1/4√55.
<0,200> = T1 <1/4√55, 3/4> + T2 <-1/4√7, 3/4>
0 = T1 * 1/4√55 - T2 * 1/4√7
200 = T1 * 3/4 + T2 * 3/4
Solving system of equations:
T1 = 70.11898262
T2 = 196.5476841
It seems right, and this is how I would done it in physics but the notion of "magnitude" rather it applies to the tension or the length of the string in this case is really throwing me off, and if I have to normalize the vector by dividing it by it's magnitude, and which magnitude? tension or length? I believe learning vectors in physics first messes up my calculus understanding, or my teacher confused me.