How to Calculate Work Done with Vectors?

[LondonLady]

Hello again

I have another question!

Suppose a particles initial position is $$\vec{r_1} = 2\vec{i} + 5\vec{j} - \vec{k}$$ metres and its acted upon by a force $$\vec{F} = \vec{i} + \vec{j} + \vec{k}$$ Newtons. Its final position is $$\vec{r_2} = -4\vec{i} + 3\vec{j} + \vec{k}$$. Find the work done by $$\vec{F}$$.

Ok, i have the formula $$dW = \vec{F}.d\vec{r}$$ joules.

what is dr? is it simply the change in the position vector? How do I start this off?

Thankyou,

 

[Pyrrhus]

It looks to me like a

$$W = \int_{\vec{r}_{o}}^{\vec{r}} \vec{F} \cdot d \vec{r}$$

Use the components

$$W = \int_{(r_{x_{o}},r_{y_{o}},r_{z_{o}})}^{(r_{x},r_{y},r_{z})}} (F_{x} \vec{i} + F_{y} \vec{j} + F_{z} \vec{k}) \cdot (dr_{x} \vec{i} + dr_{y} \vec{j} + dr_{z} \vec{k})$$

 

[LondonLady]

Thankyou very much