# Two vector ( vec{A} ) and ( vec{B} ) have equal magnitude. The

magnitude of ( (vec{A}+vec{B}) ) is ( n ) times the magnitude of

( (vec{A}-vec{B}) . ) The angle between ( vec{A} ) and ( vec{B} ) is:

where " ( theta ) is angle between vector

( |A-B|=sqrt{A^{2}+B^{2}-2 A B cos theta}={text { }}{} )

[

|A+B|=sqrt{A^{2}+B^{2}+2 A B cos theta}=

]

as given A, B have same magnitudes

[

begin{array}{c}

(A-B)=sqrt{2 A^{2}-2 A^{2} cos theta}

(A+B)=sqrt{2 A^{2}+2 A^{2} cos theta}

qquad begin{array}{c}

g_{i v e n} cdot nleft|A-B|_{}=right| A+B mid

n sqrt{2 A^{2}-2 A^{2} cos theta}=sqrt{2 A^{2}+2 A^{2} cos theta}

end{array}

end{array}

]

Squaring on both sides

[

begin{array}{c}

n^{2}left(2 A^{2}-2 A^{2} cos thetaright)=left(2 A^{2}+2 A^{2} cos thetaright) ^{2} times 2A^{2}(1 - cos theta) = 2A^{2}(1 + cos theta)

n^{2}-n^{2} cos theta=1+cos theta

n^{2}-1=cos thetaleft(1+n^{2}right)

cos theta=frac{n^{2}-1}{n^{2}+1}

theta=cos ^{2}left(frac{n^{2}-1}{n^{2}+1}right)

end{array}

]