Let `vec A(t) = f_1(t) hat i + f_2(t) hat j and vec B(t) = g(t)hat i+g_2(t) hat j,t in [0,1],f_1,f_2,g_1 g_2` are continuous functions. If `vec A(t) and vec B(t)` are non-zero vectors for all `t and vec A(0) = 2hat i + 3hat j,vec A(1) = 6hat i + 2hat j, vec B(0) = 3hat i + 2hat i and vec B(1) = 2hat j + 6hat j` Then,show that `vec A(t) and vec B(t)` are parallel for some `t`.

Explanation

If A(t) and B(t) are non-zero vectors for all t
and A(0)=`2hati+3hatj,A(1)=6hati+2hatj,B(0)=3hati+2hatj`,
and `B(1)=2hati+6hatj`
In order to prove that `A(t) and B(t)` are parallel vectors for some values of t. it is sufficient to show A(t)=`lamdaB(t)` for some `lamda`.
`hArr {f_(1)(t)hati+f_(2)(t)hatj}=lamda{g_(1)(t)hati+g_(2)(t) hatj}`
`hArr f_(1)(t)=lamdag_(1)t and f_(2)(t)=lamdag_(2)(t)`
`harr(f_(1)(t))/(f_(2)(t))=(g_(1)(t))/(g_(2)(t))`
`harr f_(2)(t)g_(2)(t)-f_(2)(t)g_(1)t=0` for some `t in [0,1]`
let `f(t)=f_(1)(t)g_(2)(t)-f_(2)(t)g_(1)(t),t in [0,1]`
Since, `f_(1),f_(2),g_(1) and g_(2)` are continuous functions.
`thereforeF(t)` is also a continuous functions.
Also, `f(0)=f_(1)(0)g_(2)(0)-g_(1)(0)f_(2)(0)`
`=2xx2-3xx3=4-9=-5 lt 0`
and `f(1)=f_(1)(1)g_(2)(1)-g_(1)(1)f_(2)(1)`
`=6xx6-2xx2=32 gt 0`
thus, F(t) is a continuous function on [0,1] such that `F(0)*F(1) lt 0`.
`therefore`By intermediate value theorem, there exists some `t in (0,1)`
such that
`f(t)=0`
`implies f(t)g_(2)(t)-f_(2)(t)g_(1)t=0`
`implies A(t)=lamdaB(t)` for some `lamda`.
Hence, A(t) and B(t) are parallel vectors.