Generating...                               quiz02a_n28

  1. Find the line of intersection of the planes −2x + 2y − z = 0 and 9 − x + 3y + z = 0 using the point [  − 2 ,   − 3 ,   − 2 ] of intersection.

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

  2. Find the tangent line to $\mathbf{r}(s) = \left[ s^2 , s , s^3 \right] $ at $s = 1$.

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

  3. Find the distance between the planes 12 − x + 3y − 3z = 0 and −5 − x + 3y − 3z = 0.

    $\displaystyle \frac{20}{\sqrt{26}}$ $\displaystyle \frac{19}{\sqrt{26}}$ $\displaystyle \frac{17}{\sqrt{19}}$ $\displaystyle \frac{18}{\sqrt{19}}$

  4. Find the line through the point [  − 2 ,  1 ,  0 ] and parallel to the vector [ 2 ,   − 2 ,   − 1 ].

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

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

    13 11 11 14

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

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

  7. Find the plane through the point [  − 4 ,  1 ,  0 ] and perpendicular to the vector [ 5 ,  4 ,  1 ].

    15 + 5x + 5y + z = 0 16 + 5x + 4y + z = 0 11 + 5x + 4y + z = 0 10 + 5x + 5y + z = 0

  8. Find the distance between the point [ 3 ,   − 1 ,   − 3 ] and the planes 2 + 2x + 4y − 2z = 0.

    $\displaystyle \frac{5}{\sqrt{6}}$ $\displaystyle \frac{11}{2\,\sqrt{6}}$ $\displaystyle \frac{7}{\sqrt{33}}$ $\displaystyle \frac{8}{\sqrt{33}}$

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

    [ 0 ,  2 ,   − 7 ] [  − 2 ,  3 ,   − 7 ] [ 0 ,  3 ,   − 6 ] [  − 2 ,  2 ,   − 8 ]

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

    −5 + 11y − 9z = 0 −10 + 10y − 10z = 0 −5 + 11y − 9z = 0 −10 + 10y − 10z = 0


Department of Mathematics
Last modified: 2026-05-20