1. Find the distance between the planes −2 − x + 2y − 3z = 0 and 9 − x + 2y − 3z = 0.

    $\displaystyle \frac{10}{\sqrt{14}}$ $\displaystyle \frac{3}{\sqrt{14}}$ $\displaystyle \frac{11}{\sqrt{14}}$ $\displaystyle \frac{4}{\sqrt{14}}$

  2. Find the plane through the points [ 3 ,  0 ,  1 ], [ 5 ,   − 2 ,  3 ], and [ 5 ,  0 ,  2 ].

    1 − 2x + 3y + 5z = 0 3 − 2x + 3y + 5z = 0 4 − 2x + 2y + 4z = 0 2 − 2x + 2y + 4z = 0

  3. Find the line of intersection of the planes −3 + 3x − y − 2z = 0 and −1 + 3x + y − 2z = 0 using the point [ 0 ,   − 1 ,   − 1 ] of intersection.

    [ x ,  y ,  z ]  = [ 4t ,   − 1 + t ,   − 1 + 7t ] [ x ,  y ,  z ]  = [ 4t ,   − 1 ,   − 1 + 6t ] [ x ,  y ,  z ]  = [ 3t ,   − 1 + t ,   − 1 − 2t ] [ x ,  y ,  z ]  = [ 3t ,   − 1 − t ,   − 1 − 2t ]

  4. Find the line through the point [  − 2 ,   − 1 ,  3 ] and parallel to the vector [  − 2 ,  2 ,  0 ].

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

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

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

  6. Find the plane through the point [ 0 ,   − 3 ,  1 ] and containing the line $\displaystyle \left[ x , y , z \right] =\left[ -2+4\,t , 1-2\,t , -3+3\,t \right] $.

    22 − 4x + 10y + 12z = 0 18 − 4x + 10y + 12z = 0 21 − 4x + 11y + 12z = 0 25 − 4x + 11y + 12z = 0

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

    [  − 5 ,  0 ,  0 ] [  − 3 ,  0 ,   − 2 ] [  − 6 ,  3 ,  1 ] [  − 4 ,  3 ,   − 1 ]

  8. Find the plane through the point [  − 4 ,   − 3 ,   − 4 ] and perpendicular to the vector [ 3 ,  4 ,   − 3 ].

    9 + 3x + 4y − 3z = 0 15 + 3x + 5y − 3z = 0 12 + 3x + 4y − 3z = 0 12 + 3x + 5y − 3z = 0

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

    2 −1 0 1

  10. Find the distance between the point [ 4 ,   − 3 ,   − 4 ] and the planes 7 + 4x − 3y − 3z = 0.

    $\displaystyle \frac{44}{\sqrt{34}}$ $\displaystyle \frac{41}{2\,\sqrt{6}}$ $\displaystyle \frac{45}{\sqrt{34}}$ $\displaystyle \frac{20}{\sqrt{6}}$


Department of Mathematics
Last modified: 2026-07-06