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Question
ABCD is a parallelogram with AC and BD as diagonals.
Then, \[\overrightarrow{AC} - \overrightarrow{BD} =\]
Options
\[4 \overrightarrow{AB}\]
- \[3 \overrightarrow{AB}\]
- \[2 \overrightarrow{AB}\]
- \[\overrightarrow{AB}\]
Solution
\[2 \overrightarrow{AB}\]
Given: ABCD, a parallelogram with diagonals AC and BD.
Then,
\[\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC} \]
\[ \overrightarrow{AD} = \overrightarrow{AB} + \overrightarrow{BD} \]
\[ \Rightarrow \overrightarrow{BD} = \overrightarrow{AD} - \overrightarrow{AB}\]
∴ \[\overrightarrow{AC} - \overrightarrow{BD} = \overrightarrow{AB} + \overrightarrow{BC} - \overrightarrow{AD} + \overrightarrow{AB} = 2 \overrightarrow{AB}\] [∵ \[\overrightarrow{AD} = \overrightarrow{BC}\]]
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