How to Calculate Work Done with Vectors?

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To calculate the work done by a force vector on a particle, the formula dW = \vec{F} · d\vec{r} is used, where d\vec{r} represents the change in the position vector. The initial and final positions of the particle are given, allowing for the determination of d\vec{r} as the difference between these two vectors. The work can be expressed as W = ∫(from r_o to r) of (F_x i + F_y j + F_z k) · (dr_x i + dr_y j + dr_z k). This integral simplifies the calculation by breaking down the components of the force and displacement. Understanding these vector components is crucial for accurately calculating the work done.
LondonLady
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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,
 
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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})
 
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Thankyou very much
 
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