|c|^2 = (a+tb)。 (a+tb)

= |a|^2+t^2|b|^2+ 2t|a||b|cosθ

d(|c|^2)/dt = 2t|b|^2+2 | a | | b | cosθ= 0

t =-|a|cosθ/ |b|

(|c|^2)'' =2|b|^2 & gt； 0 (minimum)

Min |c| when t = |a|cosθ/ |b|

=|a|sinθ

t = -|a|cosθ/ |b|

c = a -[ |a|cosθ/ |b|] b

c.b = (a -[ |a|cosθ/ |b|] b)。 b

= a.b - |a||b|cosθ =0

The included angle between b and c = π/2.