1. Let $\mathbf{r}(t) = \mathbf{u}(t)\times\mathbf{v}(t)$. Suppose that $\mathbf{u}( 1) = \left[ -3 , -2 , -2 \right] $, $\mathbf{v}( 1) = \left[ -1 , 3 , -1 \right] $, $\mathbf{u}'( 1) = \left[ -1 , 0 , 2 \right] $, and $\mathbf{v}'( 1) = \left[ 2 , -1 , 0 \right] $. Then find $\mathbf{r}'( 1)$.

    [  − 8 ,   − 7 ,  3 ] [  − 10 ,   − 6 ,  3 ] [  − 10 ,   − 6 ,  2 ] [  − 8 ,   − 7 ,  4 ]

  2. Find the plane through the point [  − 3 ,  3 ,  0 ] and containing the line [ x ,  y ,  z ]  = [ 2 − 3t ,   − 2 + t ,  3t ].

    −3 + 15x + 16y + 10z = 0 −28 + 15x + 16y + 10z = 0 −25 + 15x + 15y + 10z = 0 15x + 15y + 10z = 0

  3. Find the tangent line to $\mathbf{r}(s) = \left[ s^2 , s^2 , s^3 \right] $ at $s = -1$.

    [ x ,  y ,  z ]  = [ 2 − 2t ,  1 − 2t ,   − 1 + 3t ] [ x ,  y ,  z ]  = [ 1 − 2t ,  1 − 2t ,   − 1 + 3t ] [ x ,  y ,  z ]  = [ 1 − 2t ,  1 − t ,   − 1 + 4t ] [ x ,  y ,  z ]  = [ 2 − 2t ,  1 − t ,   − 1 + 4t ]

  4. Find the distance between the planes −8 − x − 3z = 0 and 12 − x − 3z = 0.

    $\displaystyle \frac{21}{\sqrt{11}}$ $\displaystyle \frac{19}{\sqrt{10}}$ $\displaystyle 2\,\sqrt{10}$ $\displaystyle 2\,\sqrt{11}$

  5. Find the plane through the points [  − 1 ,  1 ,  3 ], [  − 3 ,  3 ,  5 ], and [  − 1 ,  2 ,  2 ].

    3 − 4x − y − 2z = 0 4 − 4x − 2y − 2z = 0 8 − 4x − 2y − 2z = 0 7 − 4x − y − 2z = 0

  6. Let $\mathbf{r}(t) = \mathbf{u}(t)\cdot\mathbf{v}(t)$. Suppose that $\mathbf{u}( -1) = \left[ -1 , 0 , -2 \right] $, $\mathbf{v}( -1) = \left[ 1 , -3 , -3 \right] $, $\mathbf{u}'( -1) = \left[ 2 , -1 , 0 \right] $, and $\mathbf{v}'( -1) = \left[ -2 , 0 , 1 \right] $. Then find $\mathbf{r}'( -1)$.

    3 4 5 6

  7. Find the line through the point [  − 5 ,  1 ,  2 ] and perpendicular to the plane −4 − 2x − 5y − 4z = 0.

    [ x ,  y ,  z ]  = [  − 4 − 2t ,  1 − 4t ,  3 − 3t ] [ x ,  y ,  z ]  = [  − 5 − 2t ,  1 − 4t ,  2 − 3t ] [ x ,  y ,  z ]  = [  − 4 − 2t ,  1 − 5t ,  3 − 4t ] [ x ,  y ,  z ]  = [  − 5 − 2t ,  1 − 5t ,  2 − 4t ]

  8. Find the line through the points [ 3 ,   − 2 ,   − 3 ] and [ 1 ,   − 3 ,  3 ] .

    [ x ,  y ,  z ]  = [ 4 − 2t ,   − 2 ,   − 2 + 7t ] [ x ,  y ,  z ]  = [ 3 − 2t ,   − 2 ,   − 3 + 7t ] [ x ,  y ,  z ]  = [ 4 − 2t ,   − 2 − t ,   − 2 + 6t ] [ x ,  y ,  z ]  = [ 3 − 2t ,   − 2 − t ,   − 3 + 6t ]

  9. Find the line through the point [ 1 ,   − 1 ,   − 2 ] and parallel to the vector [ 4 ,   − 3 ,   − 2 ].

    [ x ,  y ,  z ]  = [ 2 + 4t ,   − 1 − 2t ,   − 2 − t ] $\displaystyle \left[ x , y , z \right] =\left[ 2+4\,t , -1-3\,t , -2-2\,t \right] $ [ x ,  y ,  z ]  = [ 1 + 4t ,   − 1 − 2t ,   − 2 − t ] $\displaystyle \left[ x , y , z \right] =\left[ 1+4\,t , -1-3\,t , -2-2\,t \right] $

  10. Find the distance between the point [  − 4 ,   − 3 ,  0 ] and the planes −17 + 4x − 3y − 4z = 0.

    $\displaystyle \frac{24}{\sqrt{41}}$ $\displaystyle \frac{23}{\sqrt{41}}$ $\displaystyle \frac{20}{\sqrt{29}}$ $\displaystyle \frac{19}{\sqrt{29}}$


Department of Mathematics
Last modified: 2026-01-04