# What is the cross product of <2 , 5 ,-7 > and <5 ,6 ,-9 >?

Feb 25, 2016

$< - 3 , - 17 , - 13 >$

#### Explanation:

{: (,,color(blue)bar(a),xx,color(blue)bar(b),=,color(blue)(barv)), (color(cyan)(x),,a_x,,b_x,,v_x), (color(cyan)(y),,a_y,,b_y,,v_y), (color(cyan)(z),,a_z,,b_z,,v_z) :}

Evaluation of the vector $\overline{v}$ can be performed in a manner similar to finding determinants.

${v}_{x} = + | \left({a}_{y} , {b}_{y}\right) , \left({a}_{z} , {b}_{z}\right) | \textcolor{w h i t e}{\text{XXX") v_y=-|(a_x,b_x),(a_z,b_z)|color(white)("XXX}} {v}_{z} = + | \left({a}_{x} , {b}_{x}\right) , \left({a}_{y} , {b}_{y}\right) |$

For the given vectors:
$\textcolor{w h i t e}{\text{XXX}} < {a}_{x} , {a}_{y} , {a}_{z} > = < 2 , 5 , - 7 >$
$\textcolor{w h i t e}{\text{XXX}} < {b}_{x} , {b}_{y} , {b}_{z} > = < 5 , 6 , - 9 >$

This becomes
{: (,,color(blue)bar(a),xx,color(blue)bar(b),=,color(blue)(barv)), (color(cyan)(x),,2,,5,,v_x), (color(cyan)(y),,5,,6,,v_y), (color(cyan)(z),,-7,,-9,,v_z) :}
and

$\overline{\textcolor{w h i t e}{\text{XXXXXXXXXX}}}$
$\textcolor{w h i t e}{\text{XXX}} {v}_{x} = + | \left(5 , 6\right) , \left(- 7 , - 9\right) |$

$\textcolor{w h i t e}{\text{XXXX}} = 5 \times \left(- 9\right) - 6 \times \left(- 7\right)$

$\textcolor{w h i t e}{\text{XXXX}} = - 45 + 42 = - 3$

$\overline{\textcolor{w h i t e}{\text{XXXXXXXXXX}}}$
$\textcolor{w h i t e}{\text{XXX}} {v}_{y} = - | \left(2 , 5\right) , \left(- 7 , - 9\right) |$

$\textcolor{w h i t e}{\text{XXXX}} = - \left(2 \times \left(- 9\right) - 5 \times \left(- 7\right)\right)$

$\textcolor{w h i t e}{\text{XXXX}} = - \left(- 18 + 35\right)$

$\textcolor{w h i t e}{\text{XXXX}} = - 17$

$\overline{\textcolor{w h i t e}{\text{XXXXXXXXXX}}}$
$\textcolor{w h i t e}{\text{XXX}} {v}_{z} = + | \left(2 , 5\right) , \left(5 , 6\right) |$

$\textcolor{w h i t e}{\text{XXXX}} = 2 \times 6 - 5 \times 5$

$\textcolor{w h i t e}{\text{XXXX}} = 12 - 25$

$\textcolor{w h i t e}{\text{XXXX}} = - 13$
$\overline{\textcolor{w h i t e}{\text{XXXXXXXXXX}}}$