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If the position vector of one end of the line segment $A B$ be $2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}$ and the position vector of its middle point be $3(\mathbf{i}+\mathbf{j}+\mathbf{k})$, then the position vector of the other end is
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Verified Answer
The correct answer is:
$4 \mathbf{i}+3 \mathbf{j}+7 \mathbf{k}$
$\overrightarrow{O A}=2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}, \quad \overrightarrow{O P}=3(\mathbf{i}+\mathbf{j}+\mathbf{k}), \quad \overrightarrow{O B}=?$
we have $\overrightarrow{O P}=\frac{\overrightarrow{O A}+\overrightarrow{O B}}{2}$
$\begin{aligned}
\Rightarrow \overrightarrow{O B} & =2 \overrightarrow{O P}-\overrightarrow{O A} \\
& =4 \mathbf{i}+3 \mathbf{j}+7 \mathbf{k}
\end{aligned}$
Trick : By inspection, middle point of $4 \mathbf{i}+3 \mathbf{j}+7 \mathbf{k}$ and $2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}$ is $3(\mathbf{i}+\mathbf{j}+\mathbf{k})$.
we have $\overrightarrow{O P}=\frac{\overrightarrow{O A}+\overrightarrow{O B}}{2}$
$\begin{aligned}
\Rightarrow \overrightarrow{O B} & =2 \overrightarrow{O P}-\overrightarrow{O A} \\
& =4 \mathbf{i}+3 \mathbf{j}+7 \mathbf{k}
\end{aligned}$
Trick : By inspection, middle point of $4 \mathbf{i}+3 \mathbf{j}+7 \mathbf{k}$ and $2 \mathbf{i}+3 \mathbf{j}-\mathbf{k}$ is $3(\mathbf{i}+\mathbf{j}+\mathbf{k})$.
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