Generating...                               quiz02a_n4

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

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

  2. Find the line of intersection of the planes 6 − 3y − z = 0 and 6 + 3x − 3y + 2z = 0 using the point [  − 3 ,  1 ,  3 ] of intersection.

    [ x ,  y ,  z ]  = [  − 3 + 3t ,  1 − 3t ,  3 + 2t ] $\displaystyle \left[ x , y , z \right] =\left[ -3-9\,t , 1-2\,t , 3+10\,t \right] $ [ x ,  y ,  z ]  = [  − 3 ,  1 − 3t ,  3 − t ] [ x ,  y ,  z ]  = [  − 3 − 9t ,  1 − 3t ,  3 + 9t ]

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

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

  4. Find the distance between the planes 10 − 3x + 3y − z = 0 and 6 − 3x + 3y − z = 0.

    $\displaystyle \frac{6}{\sqrt{26}}$ $\displaystyle \frac{4}{\sqrt{19}}$ $\displaystyle \frac{5}{\sqrt{26}}$ $\displaystyle \frac{5}{\sqrt{19}}$

  5. Find the distance between the point [  − 1 ,  1 ,  2 ] and the planes −10 − x − 3y − 3z = 0.

    $\displaystyle \frac{16}{\sqrt{14}}$ $\displaystyle \frac{15}{\sqrt{14}}$ $\displaystyle \frac{17}{\sqrt{19}}$ $\displaystyle \frac{18}{\sqrt{19}}$

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

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

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

    −20 − 18x + 14y + 13z = 0 −23 − 18x + 15y + 14z = 0 −19 − 18x + 15y + 14z = 0 −25 − 18x + 14y + 13z = 0

  8. Find the tangent line to $\mathbf{r}(s) = \left[ s^4 , s^2 , s \right] $ at $s = 1$.

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

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

    15 14 16 17

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

    [ 9 ,   − 15 ,   − 2 ] [ 5 ,   − 12 ,   − 2 ] [ 7 ,   − 15 ,   − 4 ] [ 7 ,   − 12 ,  0 ]


Department of Mathematics
Last modified: 2025-12-21