If →a=x^i+(x−1)^j+^k and →b=(x+1)^i+^j+a^k always make an acute angle with each other for every value of x ϵ R, then
a ϵ (2,∞)
→a.→b=(x^i+(x−1)^j+^k).((x+1)^i+^j+a^k)=x(x+1)+x−1+a=x2+2x+a−1
If the angle between →a and →b is acute
⇒→a.→b>0 ∀ x ϵ R⇒x2+2x+a−1>0 ∀ x ϵ R
The discriminant of the quadratic expression must be negative
⇒4−4(a−1)<0⇒a>2