Question

A constant force \(\vec{F} = (4\hat{i} + \hat{j} - 3\hat{k}) \, \text{N} \) moves a particle from \( A: (1, 2, 3) \, \text{m}  \text{to}  B: (5, 4, 1) \, \text{m}. \)
Find the work done by the force (in joules). Answer as an integer.
 

Hint:

Always form the displacement \(\vec d=\vec r_B-\vec r_A\) first, then use \(W=\vec F\cdot\vec d\). A quick sign check: if any component of \(\vec F\) and \(\vec d\) have opposite signs, their product is negative and reduces the work; same signs increase it.

Answer 1

Step 1: Displacement vector.
Position vectors: \(\vec r_A=1\hat i+2\hat j+3\hat k\), \(\vec r_B=5\hat i+4\hat j+1\hat k\).
Displacement \(\vec d=\vec r_B-\vec r_A\)
\[ \Rightarrow\ \vec d=(5-1)\hat i+(4-2)\hat j+(1-3)\hat k =4\hat i+2\hat j-2\hat k. \] 

Step 2: Work done by a constant force.
Work \(W=\vec F\cdot\vec d\) (dot product selects the component of force along displacement).
Compute componentwise: \[ \vec F\cdot\vec d =(4)(4)+(\,1)(2)+(-3)(-2) =16+2+6=24\ \text{J}. \] \[ \boxed{24} \]