Generating...                               quiz02a_n21

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

    [ x ,  y ,  z ]  = [ 2 − t ,   − 2 + 3t ,  6 − 4t ] [ x ,  y ,  z ]  = [ 1 − t ,   − 2 + 2t ,  5 − 5t ] [ x ,  y ,  z ]  = [ 1 − t ,   − 2 + 3t ,  5 − 4t ] [ x ,  y ,  z ]  = [ 2 − t ,   − 2 + 2t ,  6 − 5t ]

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

    −12 −13 −12 −15

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

    −32 + 16x + 16y + 16z = 0 −16 + 16x + 16y + 16z = 0 −18 + 16x + 17y + 17z = 0 −34 + 16x + 17y + 17z = 0

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

    −2 + 2x − 3y − z = 0 −6 + 2x − 4y − z = 0 −7 + 2x − 4y − z = 0 −3 + 2x − 3y − z = 0

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

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

    [  − 10 ,   − 3 ,  5 ] [  − 8 ,   − 3 ,  4 ] [  − 9 ,   − 3 ,  4 ] [  − 7 ,   − 3 ,  3 ]

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

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

  8. Find the distance between the planes 2x + 2y + 3z = 0 and 2 + 2x + 2y + 3z = 0.

    $\displaystyle \frac{2}{\sqrt{17}}$ $\displaystyle \frac{1}{\sqrt{17}}$ $\displaystyle \frac{3}{\sqrt{22}}$ $\displaystyle \frac{2}{\sqrt{22}}$

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

    −1 − 4x − 5y + 2z = 0 3 − 4x − 5y + 2z = 0 −4x − 6y + 2z = 0 4 − 4x − 6y + 2z = 0

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

    $\displaystyle \frac{16}{\sqrt{41}}$ $\displaystyle \frac{17}{\sqrt{41}}$ $\displaystyle \frac{29}{\sqrt{43}}$ $\displaystyle \frac{28}{\sqrt{43}}$


Department of Mathematics
Last modified: 2025-06-19