Vectors...given that a+b+c=0. Two out of the three vectors are equal in magnitude. The magnitude of the third vector if root2 times that of the either out of the other two. Find the angles b/w...

Vectors...

given that a+b+c=0. Two out of the three vectors are equal in magnitude. The magnitude of the third vector if root2 times that of the either out of the other two. Find the angles b/w these vectors.

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quantatanu | Student, Undergraduate | (Level 1) Valedictorian

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As 

a+b+c=0

=> a+b = -c

and so

|a+b| = |c|

that means resultant of vector sum of the two (a & b) is equal to -C.

Now 

let

|a|=|b|=a

given

|c|=Sqrt[2] a and is equal to |a+b|

Now resultant of vector sum of a & b is

|a+b| = Sqrt[ |a|^2 + |b|^2 + 2.|a|.|b| Cos(angle between a & b) ]

        =  Sqrt[ 2 a^2 + 2 a^2 Cos(angle between a & b) ]

=> Sqrt[2] a = Sqrt[2] a Sqrt[1+Cos(angle between a & b) ]

=> Sqrt[1+Cos(angle between a & b) ] = 1

=> 1+Cos(angle between a & b) = 1

=>  Cos(angle between a & b) = 0

=> angle between a & b = 90 degrees

and as a & b are equal in magnitude and " - c "  is the resultant of vector sum of a & b, so - c lies exactly in the middle of a & b, that is " - c" makes 45 degree angles with a & b, and hence "c" makes "180 - 45" degree angle between a & b,

so 

angle between a & b = 90 degrees

angle between a & c = angle between b & c = 180 - 45 degree

                                                              = 135 degree

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