The vectors 3a−5b and 2a+b are mutually perpendicular and the vectors a+4b and −a+b are also mutually perpendicular then the acute angle between a and b is
(3a−5b)⟂(2a+b) ⇒(3a−5b)(2a+b)=0 ⇒6a⋅a+3a⋅b−10b⋅a−5b⋅b=0 ⇒6|a|2+3a⋅b−10a⋅b−5|b|2=0 ⇒6|a|2−5|b|2=7a⋅b.....(i) Also, (a+4b)⟂(−a+b) ⇒(a+4b)⋅(−a+b)=0 ⇒−a⋅a+a⋅b−4b⋅a+4b⋅b=0 ⇒−|a|2+a⋅b−4a⋅b+4|b|2=0 ⇒−|a|2+4|b|2=3a⋅b.....(ii) From Eqs. (i) and (ii), we have ‌‌