Angle between to parametric vector

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Then the corresponding angle is the angle between two vectors which can be calculated using calculus$$v_1=r_1'=(1,2t,3t^2)\\v_2=r_2'=(\cos t,2\cos 2t,1)$$therefore $$v_1=(1,0,0)\\v_2=(1,2,1)\\\to\theta=\cos^{-1}\left(\dfrac{v_1\cdot v_2}{|v_1||v_2|}\right)=\cos^{-1}\left(\dfrac{1}{\sqrt 6}\right)$$

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xavier corbeil
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xavier corbeil

Updated on January 08, 2023

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  • xavier corbeil
    xavier corbeil 10 months

    Information given:
    $$r_{1} = (t,t^2,t^3)$$ $$r_{2} = (\sin t,\sin2t,t)$$ These vector intersect at origin $(0,0,0)$

    What I need to find: The angle between them.

    What I tried: I tried to use the formula $\theta=\cos^{-1}\left(\dfrac{v_1\cdot v_2}{|v_1||v_2|}\right)$ , where $v_1 = r_1, v_2 =r_2$. But when I calculate $v_1.v_2$ I obtain $(t\sin t+t^2\sin {2t} + t^4)$ where $t=0$ so the angle is always 0.

    The answer to this problem: around 66$^{\circ}$

  • user
    user over 5 years
    Why did you take the derivative?
  • xavier corbeil
    xavier corbeil over 5 years
    Yea I would like to know too, because this is the good answer
  • Mostafa Ayaz
    Mostafa Ayaz over 5 years
    Well! It is the tangent vector on its corresponding curve which is derived using $$\dfrac{r(x+\Delta x)-r(x)}{\Delta x}$$
  • Mostafa Ayaz
    Mostafa Ayaz over 5 years
    Also use your intuition on 2D curves