How do you find the cross product and state whether the resulting vectors are perpendicular to the given vectors $<-1,1,0>times<2,1,3>$ ?

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Answer 1 · 2017-01-04T13:33:57

$vecaxxvecb=+3hatveci+3hatvecj-3hatveck$

The vector cross product of two vectors $veca" "$ & $" "vecb" "$

is defined as

$vecaxxvecb=|veca||vecb|sinthetahatvecn$

where $theta" "$ is the angle between the vectors & #" "hatvecn" "isa unit vector mutually perpendicular to

$veca" "$ & $" "vecb" "$

If the vectors are in component form, in particular:

$veca=((a_1),(a_2),(a_3))$ and $vecb=((b_1),(b_2),(b_3))$

we can evaluate the cross product by the use of a determinate

$vecaxxvecb=|(hatveci,hatvecj,hatveck),(a_1,a_2, a_3),(b_1,b_2,b_3)|$

In this case we have

$veca=((-1),(1),(0))$ $" "vecb=((2),(1),(3))$

$vecaxxvecb=|(hatveci,hatvecj,hatveck),(-1,1,0),(2,1,3)|$

expanding the determinant as usual

$vecaxxvecb=+hatveci|(1,0),(1,3)|-hatvecj|(-1,0),(2,3)|+hatveck|(-1,1),(2,1)|$

$vecaxxvecb=+3hatveci+3hatvecj-3hatveck$

By definition this is perpendicular tot eh original two vectors. A quick check using the dot product will confirm this.

$veca.((3),(3),(-3))=((-1),(1),(0)).((3),(3),(-3))$

$=-3+3+0=0" "$ perpendicular

$vecb.((3),(3),(-3))=((2),(1),(3)).((3),(3),(-3))$

$=6+3-3=0" "$ perpendicular

How do you find the cross product and state whether the resulting vectors are perpendicular to the given vectors <-1,1,0>times<2,1,3><-1,1,0>times<2,1,3>?