1. Find the plane through the point [  − 2 ,   − 3 ,   − 2 ] and containing the line $\displaystyle \left[ x , y , z \right] =\left[ -1+3\,t , 3-6\,t , -1+2\,t \right] $.

    14 − 18x + 25z = 0 7 − 18x + 25z = 0 9 − 18x − y + 24z = 0 3 − 18x − y + 24z = 0

  2. Find the distance between the planes 8 − 2y + 2z = 0 and −12 − 2y + 2z = 0.

    $\displaystyle \frac{21}{2^{\frac{3}{2}}}$ $\displaystyle 5\,\sqrt{2}$ $\displaystyle \frac{23}{\sqrt{10}}$ $\displaystyle \frac{22}{\sqrt{10}}$

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

    [ 7 ,  11 ,  3 ] [ 4 ,  9 ,  4 ] [ 7 ,  11 ,  2 ] [ 4 ,  9 ,  5 ]

  4. Find the line through the points [  − 2 ,  2 ,  0 ] and [  − 3 ,  1 ,   − 1 ] .

    [ x ,  y ,  z ]  = [  − 2 − t ,  2 − t ,   − t ] [ x ,  y ,  z ]  = [  − 2 − t ,  2 ,   − t ] [ x ,  y ,  z ]  = [  − 1 − t ,  2 ,  1 − t ] [ x ,  y ,  z ]  = [  − 1 − t ,  2 − t ,  1 − t ]

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

    5 6 3 6

  6. Find the line through the point [ 3 ,   − 3 ,   − 4 ] and perpendicular to the plane 1 − 3y + 5z = 0.

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

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

    −5 − x + 2z = 0 −12 − x + y + 3z = 0 −6 − x + 2z = 0 −10 − x + y + 3z = 0

  8. Find the line through the point [  − 3 ,  1 ,   − 3 ] and parallel to the vector [ 0 ,  0 ,  1 ].

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

  9. Find the distance between the point [  − 4 ,  2 ,   − 2 ] and the planes 11 − 2x − 3y − 4z = 0.

    $\displaystyle \sqrt{17}$ $\displaystyle \frac{22}{\sqrt{29}}$ $\displaystyle \frac{21}{\sqrt{29}}$ $\displaystyle \frac{18}{\sqrt{17}}$

  10. Find the plane through the point [  − 5 ,  5 ,   − 4 ] and perpendicular to the vector [ 1 ,  1 ,   − 2 ].

    −9 + x + y − 2z = 0 −14 + x + 2y − 2z = 0 −8 + x + y − 2z = 0 −13 + x + 2y − 2z = 0


Department of Mathematics
Last modified: 2025-11-04