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

    [ 20 ,   − 8 ,  26 ] [ 20 ,   − 11 ,  26 ] [ 16 ,   − 8 ,  26 ] [ 16 ,   − 11 ,  26 ]

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

    −8 −9 −7 −10

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

    [ 1 ,   − 1 ,  1 ] $\displaystyle \left[ \frac{40}{41} , -\frac{30}{41} , \frac{40}{41} \right] $ $\displaystyle \left[ \frac{36}{41} , -\frac{27}{41} , \frac{36}{41} \right] $ [ 1 ,   − 1 ,  1 ]

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

    $\displaystyle -\frac{1}{\sqrt{5}}$ 0 $\displaystyle \frac{2}{\sqrt{13}}$ 0

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

    [  − 2 ,   − 24 ,   − 10 ] [  − 6 ,   − 19 ,   − 5 ] [  − 6 ,   − 24 ,   − 6 ] [  − 2 ,   − 19 ,   − 8 ]

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

    $\displaystyle$    not orthogonal $\displaystyle$    orthogonal

  7. $\mathbf{a}$ = [ 0 , -3 , -1 ] , $\mathbf{b}$ = [ -1 , 1 , 3 ] , and $\mathbf{c}$ = [ -11 , -17 , -1 ] are

    $\displaystyle$    coplanar $\displaystyle$    not coplanar

  8. Let $\displaystyle z^2+6\,z+y^2+x^2+10\,x+33=0$ . Find center and radius for the equation of sphere.

    The center is [  − 5 ,  0 ,   − 3 ] and the radius is 2 The center is [  − 5 ,  1 ,   − 3 ] and the radius is 1 The center is [  − 5 ,  0 ,   − 2 ] and the radius is 2 The center is [  − 5 ,  1 ,   − 3 ] and the radius is 2 The center is [  − 5 ,  0 ,   − 2 ] and the radius is 1 The center is [  − 5 ,  0 ,   − 3 ] and the radius is 1

  9. Let $\mathbf{a} =$ [ -2 , 1 , 3 ] . Find $\vert\mathbf{a}\vert$.

    $\displaystyle \sqrt{14}$ $\displaystyle \sqrt{17}$ $\displaystyle \sqrt{11}$ $\displaystyle \sqrt{21}$

  10. Find the unit vector in the direction of [ -3 , 3 , -3 ] .

    $\displaystyle \left[ -\frac{3}{\sqrt{34}} , \frac{3}{\sqrt{34}} , -\frac{2}{ \sqrt{34}} \right] $ $\displaystyle \left[ -\frac{3}{\sqrt{34}} , \frac{4}{\sqrt{34}} , -\frac{3}{ \sqrt{34}} \right] $ $\displaystyle \left[ -\frac{1}{\sqrt{3}} , \frac{1}{\sqrt{3}} , -\frac{1}{\sqrt{3 }} \right] $ $\displaystyle \left[ -\frac{2}{\sqrt{22}} , \frac{3}{\sqrt{22}} , -\frac{3}{ \sqrt{22}} \right] $

  11. Suppose that $\vert\mathbf{a}\vert = 2$ and $\vert\mathbf{b}\vert = 5$, and that $\theta = \frac{2\,\pi}{3}$ is the angle between $\mathbf{a}$ and $\mathbf{b}$. Find $\mathbf{a}\cdot\mathbf{b}$.

    −5 $\displaystyle 5\,2^{\frac{3}{2}}$ −10 $\displaystyle 5\,\sqrt{2}$

  12. Suppose that $\vert\mathbf{a}\vert = 4$ and $\vert\mathbf{b}\vert = 3$, and that $\theta = \frac{\pi}{2}$ is the angle between $\mathbf{a}$ and $\mathbf{b}$. Find $\vert\mathbf{a}\times\mathbf{b}\vert$.

    $\displaystyle 3\,2^{\frac{3}{2}}$ 18 12 $\displaystyle 9\,\sqrt{2}$

  13. Let A [ -3 , -4 , -2 ] , B [ -2 , -2 , 0 ] , and C [ -5 , -9 , -6 ] . Find the area of triangle ABC.

    $\displaystyle \frac{\sqrt{5}}{2}$ $\displaystyle 2^{\frac{3}{2}}$ $\displaystyle \sqrt{14}$ 0


Department of Mathematics
Last modified: 2026-06-21