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If $\mathbf{i}+\mathbf{j}-\mathbf{k}$ and $2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}$ are adjacent sides of a parallelogram, then the lengths of its diagonals are
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Verified Answer
The correct answer is:
$\sqrt{21}, \sqrt{13}$
$\begin{array}{ll}\text { Let } & \mathbf{A B}=\mathbf{i}+\mathbf{j}-\mathbf{k} \\ \text { and } & \mathbf{B C}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}\end{array}$
The diagonal $\mathbf{A C}=\mathbf{A B}+\mathbf{B C}$
$$
\begin{aligned}
&=(\mathbf{i}+\mathbf{j}-\mathbf{k})+(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}) \\
&=3 \mathbf{i}-2 \mathbf{j}
\end{aligned}
$$
The length of diagonal
$$
\mathbf{A C}=\sqrt{(3)^{2}+(-2)^{2}}=\sqrt{9+4}=\sqrt{13}
$$
The diagonal $\mathbf{D B}=\mathbf{A B}-\mathbf{A D}$
$$
\begin{aligned}
&=(\mathbf{i}+\mathbf{j}-\mathbf{k})-(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}) \\
&=-\mathbf{i}+4 \mathbf{j}-2 \mathbf{k}
\end{aligned}
$$
The length of diagonal
$$
\text { DB }=\sqrt{1+16+4}=\sqrt{21}
$$
The diagonal $\mathbf{A C}=\mathbf{A B}+\mathbf{B C}$
$$
\begin{aligned}
&=(\mathbf{i}+\mathbf{j}-\mathbf{k})+(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}) \\
&=3 \mathbf{i}-2 \mathbf{j}
\end{aligned}
$$
The length of diagonal
$$
\mathbf{A C}=\sqrt{(3)^{2}+(-2)^{2}}=\sqrt{9+4}=\sqrt{13}
$$
The diagonal $\mathbf{D B}=\mathbf{A B}-\mathbf{A D}$
$$
\begin{aligned}
&=(\mathbf{i}+\mathbf{j}-\mathbf{k})-(2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}) \\
&=-\mathbf{i}+4 \mathbf{j}-2 \mathbf{k}
\end{aligned}
$$
The length of diagonal
$$
\text { DB }=\sqrt{1+16+4}=\sqrt{21}
$$
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