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Question
ABCD is a parallelogram. If the coordinates of A, B, C are (−2, −1), (3, 0) and (1, −2) respectively, find the coordinates of D.
Solution
Let the coordinates of D is (x, y).
ABCD is a parallelogram.
AB = DC
We have,
\[\overrightarrow{AB} = \overrightarrow{DC} \]
\[ \Rightarrow 3 \stackrel\frown{i} - \left( - 2 \stackrel\frown{i} - \stackrel\frown{j} \right) = \left( \stackrel\frown{i} - 2 \stackrel\frown{j} \right) - \left( x \stackrel\frown{i} + y \stackrel\frown{j} \right)\]
\[ \Rightarrow 5 \stackrel\frown{i} + \stackrel\frown{j} = \stackrel\frown{i} \left( 1 - x \right) + \stackrel\frown{j} \left( - 2 - y \right)\]
\[ \Rightarrow 1 - x = 5\text{ and }1 = - 2 - y\]
\[ \Rightarrow x = - 4\text{ and }y = - 3\]
Hence, the coordinates of D is \[\left( - 4, - 3 \right)\]
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