Find all points of intersection between the graphs of the vector functions , and find the acute angle of intersection of the curves at those points.

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Consider and .

The objective is to find all point of intersection of $r_1\left(t\right)$and $r_2\left(t\right)$at this point of intersection, the angle between the curves $r_1\left(t\right)$and $r_2\left(t\right)$is to determined .

and

and

and

Thus the curves$r_1\left(r\right)$and$r_2\left(t\right)$intersect at$(0,0)$and$(1,1)$.

the tangent vectors to $r_1\left(t\right)$and $r_2\left(t\right)$are

and

The tangent vectors when $t=0$are and

Let and

The cosine of the angle $\theta$between $r_1\left(t\right)$and $r_2\left(t\right)$is

Thus at $(0,0)$, the angle of intersection between the two curves is${90}^o$

The tangent vectors when $t=1$are and .

Let and .

The cosine of the angle $\theta \text{'}$between $r_1\left(t\right)$and $r_2\left(t\right)$is

Thus at $(1,1)$, the angle of intersection between the two curves is${\cos }^{-1}\left(\frac{4}{5}\right)$