Generating...                               quiz01_n17

  1. $\mathbf{a}$ = [ 1 , -2 , 5 ] and $\mathbf{b}$ = [ 2 , 18 , 7 ] are

    $\displaystyle$    orthogonal $\displaystyle$    not orthogonal

  2. Suppose that $\vert\mathbf{a}\vert = 1$ and $\vert\mathbf{b}\vert = 4$, and that $\theta = \frac{\pi}{4}$ is the angle between $\mathbf{a}$ and $\mathbf{b}$. Find $\mathbf{a}\cdot\mathbf{b}$.

    $\displaystyle 2^{\frac{3}{2}}$ $\displaystyle 3\,2^{\frac{3}{2}}$ $\displaystyle 2\,3^{\frac{3}{2}}$ $\displaystyle 2\,\sqrt{3}$

  3. Let A [ -2 , -3 , 2 ] , B [ -4 , -5 , 5 ] , and C [ -7 , -3 , -1 ] . Find the area of triangle ABC.

    $\displaystyle \sqrt{115}$ $\displaystyle \frac{3\,\sqrt{66}}{2}$ $\displaystyle \frac{\sqrt{454}}{2}$ $\displaystyle \frac{\sqrt{577}}{2}$

  4. Find the unit vector in the direction of [ 1 , 0 , 2 ] .

    $\displaystyle \left[ \frac{1}{\sqrt{6}} , \frac{1}{\sqrt{6}} , \frac{2}{\sqrt{6}} \right] $ $\displaystyle \left[ \frac{1}{\sqrt{5}} , 0 , \frac{2}{\sqrt{5}} \right] $ $\displaystyle \left[ \frac{1}{\sqrt{2}} , 0 , \frac{1}{\sqrt{2}} \right] $ $\displaystyle \left[ \frac{1}{\sqrt{6}} , 0 , \frac{3}{\sqrt{6}} \right] $

  5. Suppose that $\vert\mathbf{a}\vert = 5$ and $\vert\mathbf{b}\vert = 2$, and that $\theta = \frac{\pi}{6}$ is the angle between $\mathbf{a}$ and $\mathbf{b}$. Find $\vert\mathbf{a}\times\mathbf{b}\vert$.

    5 $\displaystyle 5\,\sqrt{2}$ $\displaystyle 7\,\sqrt{2}$ 7

  6. Let $\mathbf{a}$ = [ 2 , -4 , 1 ] and $\mathbf{b}$ = [ 1 , -5 , -3 ] . Find comp$_{\mathbf{a}}\mathbf{b}$.

    $\displaystyle \frac{16}{\sqrt{29}}$ $\displaystyle \frac{19}{\sqrt{21}}$ $\displaystyle \frac{17}{\sqrt{29}}$ $\displaystyle \frac{17}{\sqrt{21}}$

  7. Let $\displaystyle z^2-4\,z+y^2+8\,y+x^2+8\,x+20=0$ . Find center and radius for the equation of sphere.

    The center is [  − 4 ,   − 3 ,  2 ] and the radius is 4 The center is [  − 4 ,   − 4 ,  2 ] and the radius is 5 The center is [  − 4 ,   − 4 ,  3 ] and the radius is 4 The center is [  − 4 ,   − 4 ,  2 ] and the radius is 4 The center is [  − 4 ,   − 3 ,  2 ] and the radius is 5 The center is [  − 4 ,   − 4 ,  3 ] and the radius is 5

  8. Let $\mathbf{a}$ = [ 3 , -1 , 2 ] and $\mathbf{b}$ = [ -1 , -1 , -2 ] . Find proj$_{\mathbf{a}}\mathbf{b}$.

    $\displaystyle \left[ -\frac{12}{13} , \frac{3}{13} , -\frac{9}{13} \right] $ $\displaystyle \left[ -\frac{6}{7} , \frac{2}{7} , -\frac{4}{7} \right] $ $\displaystyle \left[ -\frac{9}{7} , \frac{3}{7} , -\frac{6}{7} \right] $ $\displaystyle \left[ -\frac{18}{13} , \frac{9}{26} , -\frac{27}{26} \right] $

  9. Let $\mathbf{a}$ = [ 5 , 0 , 3 ] and $\mathbf{b}$ = [ -1 , -4 , 3 ] . Find $-2 \mathbf{a} + 4 \mathbf{b}$.

    [  − 14 ,   − 16 ,  6 ] [  − 14 ,   − 12 ,  6 ] [  − 16 ,   − 16 ,  6 ] [  − 16 ,   − 12 ,  6 ]

  10. Let $\mathbf{a}$ = [ -3 , 4 , -2 ] and $\mathbf{b}$ = [ 1 , 3 , -3 ] . Find ${\mathbf{a}}\times\mathbf{b}$.

    [  − 4 ,   − 11 ,   − 16 ] [  − 6 ,   − 8 ,   − 10 ] [  − 4 ,   − 8 ,   − 12 ] [  − 6 ,   − 11 ,   − 13 ]


Department of Mathematics
Last modified: 2025-09-14