Multiplying Vectors: Solving a.b = (a x db/dt) + (da/dt x b)

[Preciouspearl]

If a = t^2 i - (4-t)j
and b = i + t j
show d/dt (a.b) = (a x db/dt) + (da/dt x b)I know you have to multiply the vectors a and b
then do da/dt
then db/dt
and times db/dt with a
and times da/dt with b
that should be the proof

However, I don't know how to multiply the vectors!
Can someone please tell me how to multiply them

Thank you, in advance

 

[elibj123]

There are two products defined over three dimensional vectors.

First is the inner (\dot\scalar) product, that for any a=(a1,a2,a3), b=(b1,b2,b3)
Gives

a.b=a1*b1+a2*b2+a3*b3.

Which means, multiply the vector component-wise and then sum up the results. This product gives a number!

The second is the vector (\cross) product

which is:

\vec{a} X \vec{b}=(a_{2}b_{3}-b_{2}a_{3})\hat{i}+(a_{3}b_{1}-a_{1}b_{3})\hat{j}+(a_{1}b_{2}-a_{2}b_{1})\hat{k}

The result is a vector!

Now please be careful. The identity which you are trying to prove involves only dot products. So it's:

\frac{d}{dt}(\vec{a}.\vec{b})=\vec{a}.\frac{d}{dt}\vec{b}+\vec{b}.\frac{d}{dt}\vec{a}