Generating...                               quiz02a_n0

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

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

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

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

  3. Find the line of intersection of the planes −6 − 2x − y + 2z = 0 and −6 − x − 3y = 0 using the point [ 0 ,   − 2 ,  2 ] of intersection.

    [ x ,  y ,  z ]  = [ 6t ,   − 2 − 2t ,  2 + 5t ] [ x ,  y ,  z ]  = [  − 2t ,   − 2 − t ,  2 + 2t ] [ x ,  y ,  z ]  = [ 6t ,   − 2 − t ,  2 + 6t ] [ x ,  y ,  z ]  = [  − t ,   − 2 − 3t ,  2 ]

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

    8 6 9 7

  5. Find the plane through the point [  − 1 ,   − 1 ,  1 ] and containing the line [ x ,  y ,  z ]  = [  − 2 + 3t ,   − t ,   − 3 + t ].

    6 + 3x + 12y + 3z = 0 12 + 3x + 12y + 3z = 0 12 + 3x + 11y + 2z = 0 7 + 3x + 11y + 2z = 0

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

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

  7. Find the distance between the point [ 2 ,   − 3 ,  0 ] and the planes −8 + 2x − 4y + 3z = 0.

    $\displaystyle \frac{8}{\sqrt{22}}$ $\displaystyle \frac{8}{\sqrt{29}}$ $\displaystyle \frac{9}{\sqrt{29}}$ $\displaystyle \frac{7}{\sqrt{22}}$

  8. Find the plane through the points [ 2 ,  3 ,  0 ], [ 4 ,  5 ,  2 ], and [ 1 ,  5 ,   − 2 ].

    18 − 8x + 2y + 6z = 0 7 − 8x + 3y + 6z = 0 10 − 8x + 2y + 6z = 0 15 − 8x + 3y + 6z = 0

  9. Find the distance between the planes −3 − 2x − 3y = 0 and −7 − 2x − 3y = 0.

    $\displaystyle \frac{5}{2^{\frac{3}{2}}}$ $\displaystyle \frac{4}{\sqrt{13}}$ $\displaystyle \sqrt{2}$ $\displaystyle \frac{5}{\sqrt{13}}$

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

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


Department of Mathematics
Last modified: 2026-09-10