Question

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

Answer 1

`< -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 `barv` can be performed in a manner similar to finding determinants.

`v_x=+|(a_y, b_y),(a_z,b_z)|color(white)("XXX") v_y=-|(a_x,b_x),(a_z,b_z)|color(white)("XXX") v_z=+|(a_x,b_x),(a_y,b_y)|`

For the given vectors:
`color(white)("XXX")< a_x, a_y, a_z > = < 2, 5, -7 >`
`color(white)("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

`bar(color(white)("XXXXXXXXXX"))`
`color(white)("XXX")v_x=+|(5,6),(-7,-9)|`

`color(white)("XXXX")=5xx(-9)-6xx(-7)`

`color(white)("XXXX")=-45+42=-3`

`bar(color(white)("XXXXXXXXXX"))`
`color(white)("XXX")v_y=- |(2,5),(-7,-9)|`

`color(white)("XXXX")=-(2xx(-9)-5xx(-7))`

`color(white)("XXXX")=-(-18+35)`

`color(white)("XXXX")=-17`

`bar(color(white)("XXXXXXXXXX"))`
`color(white)("XXX")v_z=+|(2,5),(5,6)|`

`color(white)("XXXX")=2xx6-5xx5`

`color(white)("XXXX")=12-25`

`color(white)("XXXX")=-13`
`bar(color(white)("XXXXXXXXXX"))`