Understanding Cross Product Derivatives for Vector Functions

[kap361]

hi all.

my homework question is what is the derivative of:

[(a + t * b) x (a + t * b + t^(2) * c)]

a, b, and c are vectors, and t is a constant. * is multipy, ^(2) is squared, and x is cross product.

i've been working on it for hours and i have no idea what to do.

there's another similar problem which i have the answer to. it asks to find the derivative of:

[(a + t * b) x (c + t * d)]

it has the same specs, with d being another vector.

the answer to this one is:

(a x d) + (b x c) + 2t(b x d)

i just don't see how these connect and can actually be equivalent.

help would be ridiculously appreciated.

 

[Hurkyl]

You have a (cross) product there. So you should use the (cross) product rule.

 

[HallsofIvy]

First, the product rule, for cross product: (uxv)'= uxv'+ u'xv.

Second, cross product is anti-commutative: uxv= -vxu and, in particular, uxu= 0.
Finally, cross product is distributive: ux(v+ w)= uxv+ uxw (though is it NOT associative).

Look at your second, simpler, problem: [(a + t * b) x (c + t * d)] '
= (a+ tb)' x(c+ td)+ (a+tb)x(c+ td)'= bx(c+td)+ (a+tb)x(d)= bxc+ t(bxd)+ axd+ t(bxd)= axd+ bxc+ 2t(bxd).

 

[kap361]

ok thank you so much you two.