The curves r1(t)=⟨t,t2,t3⟩ and r2(t)=⟨sin⁡t,sin⁡2t,t⟩ intersect at the origin. Find their angle of intersection correct to the nearest degree.

Question

The curves r1(t)=⟨t,t2,t3⟩ and r2(t)=⟨sin⁡t,sin⁡2t,t⟩ intersect at the origin. Find their angle of intersection correct to the nearest degree.

Fachur · Accepted Answer

Remember that Angle between two curves at their point of intersection is the angle between their tangent vectors at that points.r1(t)=⟨t,t2,t3⟩Thereforer1′(t)=⟨1,2t,3t2⟩The parameter value corresponding to the point (0,0,0) is t=0Therefore, the tangent vector there isr1′(0)=⟨1,0,0⟩r2(t)=⟨sin⁡t,sin⁡2t,t⟩Thereforer2′(t)=⟨cos⁡t,2cos⁡2t,1⟩The parameter value corresponding to the point (0,0,0) is t=0Therefore, the tangent vector there isr2′(0)=⟨1,2,1⟩If 0 is the angle between the two curves thencos⁡0=[r1′(0)]⋅[r2′(0)]|r1′(0)||r2′(0)|    since a⋅b=|a|⋅|b|⋅cos⁡0cos⁡0=⟨1,0,0⟩⋅⟨1,2,1⟩1+0+01+4+1cos⁡0=16Therefore, angle between the two curves is cos−1(16)≈66∘Result:The angle between the two given curves is cos−1(16)≈66∘.

The curves r_{1}(t)=<< t, t^{2}, t^{3} >>\ and\ r_{2}(t)=<< \sin

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