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

    $\displaystyle$    orthogonal $\displaystyle$    not orthogonal

  2. Let $\displaystyle z^2+2\,z+y^2+x^2-2\,x+1=0$ . Find center and radius for the equation of sphere.

    The center is [ 1 ,  0 ,  0 ] and the radius is 2 The center is [ 1 ,  0 ,  0 ] and the radius is 1 The center is [ 1 ,  1 ,   − 1 ] and the radius is 1 The center is [ 1 ,  0 ,   − 1 ] and the radius is 2 The center is [ 1 ,  0 ,   − 1 ] and the radius is 1 The center is [ 1 ,  1 ,   − 1 ] and the radius is 2

  3. Let $\mathbf{a}$ = [ -2 , -2 , 3 ] and $\mathbf{b}$ = [ -5 , 0 , -3 ] . Find comp$_{\mathbf{a}}\mathbf{b}$.

    $\displaystyle -\frac{3}{\sqrt{17}}$ $\displaystyle -\frac{7}{\sqrt{21}}$ $\displaystyle -\frac{10}{\sqrt{21}}$ $\displaystyle \frac{1}{\sqrt{17}}$

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

    [ 7 ,  9 ,  2 ] [ 10 ,  13 ,  2 ] [ 7 ,  13 ,  2 ] [ 10 ,  9 ,  2 ]

  5. Find the unit vector in the direction of [ -2 , -4 , 2 ] .

    $\displaystyle \left[ -\frac{2}{\sqrt{17}} , -\frac{4}{\sqrt{17}} , \frac{3}{ \sqrt{17}} \right] $ $\displaystyle \left[ -\frac{2}{\sqrt{17}} , -\frac{3}{\sqrt{17}} , \frac{2}{ \sqrt{17}} \right] $ $\displaystyle \left[ -\frac{1}{\sqrt{6}} , -\frac{2}{\sqrt{6}} , \frac{1}{\sqrt{6 }} \right] $ $\displaystyle \left[ -\frac{1}{\sqrt{21}} , -\frac{4}{\sqrt{21}} , \frac{2}{ \sqrt{21}} \right] $

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

    13 14 9 8

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

    [ 0 ,  0 ,  0 ] $\displaystyle \left[ \frac{4}{19} , -\frac{12}{19} , -\frac{12}{19} \right] $ $\displaystyle \left[ \frac{2}{17} , -\frac{3}{17} , -\frac{2}{17} \right] $ $\displaystyle \left[ \frac{2}{19} , -\frac{6}{19} , -\frac{6}{19} \right] $

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

    [  − 2 ,   − 17 ,  9 ] [  − 2 ,   − 20 ,  10 ] [  − 3 ,   − 20 ,  5 ] [  − 3 ,   − 17 ,  5 ]

  9. Let A [ 4 , -2 , 1 ] , B [ 3 , -6 , 4 ] , and C [ -1 , -2 , -2 ] . Find the area of triangle ABC.

    $\displaystyle \frac{\sqrt{769}}{2}$ $\displaystyle \sqrt{217}$ $\displaystyle \frac{\sqrt{706}}{2}$ $\displaystyle \frac{3\,\sqrt{94}}{2}$

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

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


Department of Mathematics
Last modified: 2026-09-24