# Given |a| = 2, |b| = 5, and |a-2b| = 7.5 Determine |a+b|a and b are vectors.

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### 1 Answer

Let a be a vector such that:

a = ti + uj

==> l a l = sqrt(t^2 + u^2) = 2

==> t^2 + u^2 = 4...............(1)

b= xi + yj

==> l b l = sqrt(x^2 + y^2) = 5

==> x^2 + y^2 = 25..............(2)

a- 2b = (ti+uj) - 2( xi+yj)

a-2b = (t-2x)i + (u-2y)j

==> l a-2b l = sqrt( t-2x)^2 + (u-2y)^2 = 7.5

==> t^2 - 4tx + 4x^2 + u^2 - 4uy + 4y^2 = 7.5^2

==> t^2 + u^2 + 4x^2 + 4y^2 - 4tx - 4uy = 7.5^2

==> (t^2 + u^2) + 4 (x^2 + y^2) - 4(tx+uy) = 7.5^2

==> 4 + 4*25 - 4 (tx + uy) = 7.5^2

==> 104 - 4(tx+uy)= 7.5^2

==> -4(tx+uy) = -47.75

==> (tx+uy) = 11.9375............(3)

a+ b= ti + uj + xi+yj

= (t+x)i + (u+y)j

==, l a+ b l = sqrt(t+x)^2 + (u+y)^2

==> l a+bl = sqrt( t^2 + 2tx + x^2 + u^2 + 2uy + y^2)

==> l a+bl = sqrt( t^2+u^2 + u^2 + y^2 + 2tx+2uy)

==. l a+bl = sqrt( 4 + 25 + 2(tx+uy)

==> la+bl = sqrt( 29 + 2*11.9375)

==< la+bl = sqrt(52.875)

**==> l a+ bl = 7.272**