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

    36 − x + 2y − 5z = 0 32 − x + 3y − 4z = 0 30 − x + 2y − 5z = 0 27 − x + 3y − 4z = 0

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

    [ x ,  y ,  z ]  = [ t ,   − t ,  2 + 2t ] [ x ,  y ,  z ]  = [ 1 + t ,  0 ,  2 + 3t ] [ x ,  y ,  z ]  = [ t ,  0 ,  2 + 3t ] [ x ,  y ,  z ]  = [ 1 + t ,   − t ,  2 + 2t ]

  3. Find the line through the point [  − 2 ,   − 3 ,  2 ] and perpendicular to the plane −2 + 2y + 3z = 0.

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

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

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

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

    −8 −5 −6 −7

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

    12 + 5x + 2y − z = 0 7 + 5x + 2y − z = 0 12 + 5x + 3y = 0 7 + 5x + 3y = 0

  7. Find the distance between the planes −8 − 3x − z = 0 and 5 − 3x − z = 0.

    $\displaystyle \frac{12}{\sqrt{11}}$ $\displaystyle \sqrt{11}$ $\displaystyle \frac{13}{\sqrt{10}}$ $\displaystyle \frac{12}{\sqrt{10}}$

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

    [ 9 ,  1 ,   − 3 ] [ 6 ,  2 ,   − 4 ] [ 7 ,  3 ,   − 1 ] [ 4 ,  4 ,   − 2 ]

  9. Find the plane through the point [  − 3 ,   − 2 ,   − 2 ] and containing the line [ x ,  y ,  z ]  = [  − t ,   − 2 + 3t ,  2 − 4t ].

    6 + 12x − 7y − 8z = 0 2 + 12x − 8y − 9z = 0 −6 + 12x − 7y − 8z = 0 −10 + 12x − 8y − 9z = 0

  10. Find the line of intersection of the planes 7 + 2x + y + 3z = 0 and −4 + x − 2y + 2z = 0 using the point [  − 2 ,   − 3 ,  0 ] of intersection.

    [ x ,  y ,  z ]  = [  − 2 + 2t ,   − 3 + t ,  3t ] [ x ,  y ,  z ]  = [  − 2 + 8t ,   − 3 ,   − 5t ] [ x ,  y ,  z ]  = [  − 2 + 8t ,   − 3 − t ,   − 5t ] [ x ,  y ,  z ]  = [  − 2 + t ,   − 3 − 2t ,  2t ]


Department of Mathematics
Last modified: 2024-09-06