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Applications of Dot Product

The work done...

Question

The work done by the force $\to F = \to i + \to j + \to k$ acting on a particle is displaced from $A (3, 3, 3)$ to the point $B (4, 4, 4)$ is

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Solution

Given, $\to F = \to i + \to j + \to k$ .
Let, the position vector of $A$ is $(3, 3, 3)$ with respect to the origin $O$ and that of $B$ is $(4, 4, 4)$ .
Then, $\to O A = 3 \to i + 3 \to j + 3 \to k$ and $\to O B = 4 \to i + 4 \to j + 4 \to k$ .
Now, $\to A B =$ $\to O B$ - $\to O A = \to i + \to j + \to k$ .
$\to F$ is in the direction of displacement $\to A B$ .
So, work done $= \to F . \to A B$
$= (\to i + \to j + \to k) . (\to i + \to j + \to k) = 3$ unit.

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