A point moves from vec r_1 = (1, 1, 1) m to vec r_2 = (6, 6…

A point moves from $\vec r_1 = (1, 1, 1)\ \mathrm{m}$ to $\vec r_2 = (6, 6, 6)\ \mathrm{m}$ while a constant force $\vec F = 5\,\hat\imath\ \mathrm{N}$ acts on it. Find the work of this force.

Explanation of Method

Work of a constant force in 3D.

Step 1: Displacement

\[ \Delta\vec r = \vec r_2 - \vec r_1 = (5, 5, 5)\ \mathrm{m}. \]

Step 2: Work

\[ W = \vec F \cdot \Delta\vec r = 5 \cdot 5 + 0 \cdot 5 + 0 \cdot 5. \]

\[W = 25\ \mathrm{J}\]

$x$ $y$ $z$ $\vec r_1$ $\vec r_2$ $\vec F = 5\hat\imath\ \mathrm{N}$ $\Delta x = 5\ \mathrm{m}$ $\Delta\vec r$ $W = F\,\Delta x = 25\ \mathrm{J}$

The force points along $x$, so only the $x$-part of the displacement (5 m, red) counts; the $y$ and $z$ parts are perpendicular to the force and give no work.

Result

\[\boxed{\,W = 25\ \mathrm{J}\,}\]

Remark

Check. The original result is correct. For a constant force the work depends only on the start and end points, not on the path taken.