Generating...                               quiz02a_n19

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

    1 0 −1 0

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

    [ 3 ,  8 ,  11 ] [ 2 ,  9 ,  11 ] [ 2 ,  9 ,  10 ] [ 3 ,  8 ,  12 ]

  3. Find the plane through the point [  − 4 ,  1 ,  1 ] and perpendicular to the vector [  − 3 ,   − 4 ,   − 4 ].

    −4 − 3x − 4y − 4z = 0 −3x − 3y − 3z = 0 3 − 3x − 4y − 4z = 0 −6 − 3x − 3y − 3z = 0

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

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

  5. Find the line of intersection of the planes −8 + 2y − z = 0 and −3 − x + 2y + z = 0 using the point [ 1 ,  3 ,   − 2 ] of intersection.

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

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

    0 $\displaystyle \frac{1}{\sqrt{13}}$ 0 $\displaystyle \frac{1}{3\,\sqrt{2}}$

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

    $\displaystyle \frac{9}{\sqrt{5}}$ 3 4 $\displaystyle 2\,\sqrt{5}$

  8. Find the plane through the point [ 2 ,   − 3 ,  0 ] and containing the line [ x ,  y ,  z ]  = [ 2 − 5t ,  3 − 2t ,   − 3 ].

    −60 + 6x − 14y − 30z = 0 −54 + 6x − 14y − 30z = 0 −57 + 6x − 15y − 30z = 0 −63 + 6x − 15y − 30z = 0

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

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

  10. Find the line through the point [  − 4 ,  1 ,   − 5 ] and perpendicular to the plane −2 − 4x − 5y + z = 0.

    $\displaystyle \left[ x , y , z \right] =\left[ -4-4\,t , 1-4\,t , -5+2\,t \right] $ [ x ,  y ,  z ]  = [  − 4 − 4t ,  1 − 5t ,   − 5 + t ] $\displaystyle \left[ x , y , z \right] =\left[ -3-4\,t , 1-4\,t , -5+2\,t \right] $ [ x ,  y ,  z ]  = [  − 3 − 4t ,  1 − 5t ,   − 5 + t ]


Department of Mathematics
Last modified: 2025-09-14