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If the constant forces $2 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ and $-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}$ act on a particle due to which it is displaced from a point $A(4,-3,-2)$ to a point $B(6,1,-3)$, then the work done by the forces is
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Verified Answer
The correct answer is:
$-15$ unit
Resultant force,
$$
\overrightarrow{\mathbf{F}}=(2 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})+(-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})=\hat{\mathbf{i}}-3 \mathbf{j}+5 \hat{\mathbf{k}}
$$
Displacement,
$$
\begin{aligned}
\overrightarrow{\mathbf{d}}=\overrightarrow{\mathbf{A B}} &=(6 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})-(4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}) \\
&=2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}} \\
W &=\overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathbf{d}}=(\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}) \cdot(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}}) \\
&=2-12-5=-15 \text { unit }
\end{aligned}
$$
$$
\overrightarrow{\mathbf{F}}=(2 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})+(-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})=\hat{\mathbf{i}}-3 \mathbf{j}+5 \hat{\mathbf{k}}
$$
Displacement,
$$
\begin{aligned}
\overrightarrow{\mathbf{d}}=\overrightarrow{\mathbf{A B}} &=(6 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})-(4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}) \\
&=2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}} \\
W &=\overrightarrow{\mathbf{F}} \cdot \overrightarrow{\mathbf{d}}=(\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}) \cdot(2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-\hat{\mathbf{k}}) \\
&=2-12-5=-15 \text { unit }
\end{aligned}
$$
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