1. Find the curvature of $\displaystyle \left[ t , t^2 , 1 \right] $ at a general point.

    0 $\displaystyle \frac{2}{\left(4\,t^2+1\right)^{\frac{3}{2}}}$ 0 0

  2. Find the unit tangent vector for $\displaystyle \left[ \sin t , 4\,t , \cos t \right] $ at a general point.

    $\displaystyle \left[ \frac{\cos t}{\sqrt{17}} , \frac{4}{\sqrt{17}} , -\frac{ \sin t}{\sqrt{17}} \right] $ $\displaystyle \left[ \frac{\cos t}{\sqrt{10}} , \frac{3}{\sqrt{10}} , -\frac{ \sin t}{\sqrt{10}} \right] $ $\displaystyle \left[ \frac{2\,\cos t}{\sqrt{13}} , \frac{3}{\sqrt{13}} , -\frac{2 \,\sin t}{\sqrt{13}} \right] $ $\displaystyle \left[ \frac{\cos t}{\sqrt{5}} , \frac{2}{\sqrt{5}} , -\frac{\sin t }{\sqrt{5}} \right] $

  3. Reparametrize $\displaystyle \left[ 4\,\cos t , 2\,t , 4\,\sin t \right] $ with respect to arc length from [ 4 ,  0 ,  0 ]

    $\displaystyle \left[ 4\,\cos \left(\frac{s}{2\,\sqrt{5}}\right) , \frac{s}{\sqrt{ 5}} , 4\,\sin \left(\frac{s}{2\,\sqrt{5}}\right) \right] $ $\displaystyle \left[ \cos \left(\frac{s}{\sqrt{5}}\right) , \frac{2\,s}{\sqrt{5}} , \sin \left(\frac{s}{\sqrt{5}}\right) \right] $ $\displaystyle \left[ 4\,\cos \left(\frac{s}{\sqrt{5}}\right) , \frac{2\,s}{\sqrt{ 5}} , 4\,\sin \left(\frac{s}{\sqrt{5}}\right) \right] $ $\displaystyle \left[ \cos \left(\frac{s}{2\,\sqrt{5}}\right) , \frac{s}{\sqrt{5}} , \sin \left(\frac{s}{2\,\sqrt{5}}\right) \right] $

  4. Find the tangential component of acceleration for the position $\displaystyle \left[ 1 , t , t^3 \right] $ at a general point.

    $\displaystyle \frac{18\,t^3}{\sqrt{9\,t^4+1}}$ $\displaystyle \frac{18\,t^3+4\,t}{\sqrt{9\,t^4+4\,t^2+1}}$ $\displaystyle \frac{8\,t}{\sqrt{8\,t^2+1}}$ $\displaystyle \frac{18\,t^3}{\sqrt{9\,t^4+2}}$

  5. Find the unit tangent vector for $\displaystyle \left[ t , t^3 , t^3 \right] $ at a general point.

    $\displaystyle \left[ \frac{3\,t^2}{\sqrt{18\,t^4+4\,t^2}} , \frac{2\,t}{\sqrt{18 \,t^4+4\,t^2}} , \frac{3\,t^2}{\sqrt{18\,t^4+4\,t^2}} \right] $ $\displaystyle \left[ \frac{4\,t^3}{\sqrt{16\,t^6+8\,t^2}} , \frac{2\,t}{\sqrt{16 \,t^6+8\,t^2}} , \frac{2\,t}{\sqrt{16\,t^6+8\,t^2}} \right] $ $\displaystyle \left[ 0 , \frac{4\,t^3}{\sqrt{16\,t^6+4\,t^2}} , \frac{2\,t}{ \sqrt{16\,t^6+4\,t^2}} \right] $ $\displaystyle \left[ \frac{1}{\sqrt{18\,t^4+1}} , \frac{3\,t^2}{\sqrt{18\,t^4+1}} , \frac{3\,t^2}{\sqrt{18\,t^4+1}} \right] $

  6. Find the acceleration for the position $\displaystyle \left[ t^4 , 1 , t^3 \right] $ at a general point.

    $\displaystyle \left[ 12\,t^2 , 0 , 6\,t \right] $ [ 6t ,  6t ,  6t ] [ 6t ,  2 ,  0 ] [ 0 ,  0 ,  6t ]

  7. Find the normal component of acceleration for the position $\displaystyle \left[ \sin t , \cos t , 2\,t^2 \right] $ at a general point.

    $\displaystyle \frac{\sqrt{16\,t^2+17}}{\sqrt{16\,t^2+1}}$ $\displaystyle \frac{\sqrt{4\,t^2+5}}{\sqrt{4\,t^2+1}}$ $\displaystyle \frac{\sqrt{16\,t^2+32}}{\sqrt{4\,t^2+4}}$ $\displaystyle \frac{\sqrt{64\,t^2+80}}{\sqrt{16\,t^2+4}}$

  8. Find the curvature of $\displaystyle \left[ \cos t , \sin t , 3\,t \right] $ at a general point.

    $\displaystyle \frac{2}{13}$ $\displaystyle \frac{1}{2}$ 1 $\displaystyle \frac{1}{10}$

  9. Find the tangential component of acceleration for the position $\displaystyle \left[ 2\,t^2 , 2\,\sin t , 2\,\cos t \right] $ at a general point.

    $\displaystyle \frac{4\,t}{\sqrt{4\,t^2+1}}$ $\displaystyle \frac{16\,t}{\sqrt{16\,t^2+1}}$ $\displaystyle \frac{16\,t}{\sqrt{16\,t^2+4}}$ $\displaystyle \frac{4\,t}{\sqrt{4\,t^2+4}}$

  10. Find the binormal vector for $\displaystyle \left[ 4\,t , 2\,\cos t , 2\,\sin t \right] $ at a general point.

    $\displaystyle \left[ \frac{3}{5} , \frac{4\,\sin t}{5} , -\frac{4\,\cos t}{5} \right] $ $\displaystyle \left[ \frac{1}{\sqrt{5}} , \frac{2\,\sin t}{\sqrt{5}} , -\frac{2\, \cos t}{\sqrt{5}} \right] $ $\displaystyle \left[ \frac{3}{\sqrt{10}} , \frac{4\,\sin t}{\sqrt{10}} , -\frac{4 \,\cos t}{\sqrt{10}} \right] $ $\displaystyle \left[ \frac{2}{\sqrt{5}} , \frac{\sin t}{\sqrt{5}} , -\frac{\cos t }{\sqrt{5}} \right] $


Department of Mathematics
Last modified: 2026-06-08