Question
Let $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}},\vec{\text{d}}$ be the position vectors of the four distinct points A, B, C, D. If $\vec{\text{b}}-\vec{\text{a}}=\vec{\text{c}}-\vec{\text{d}}$, then show that ABCD is a parallelogram.
✓
Answer
Here it is given that $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}\text{ and }\vec{\text{d}}$ be the position vectors of the four distinct points A, B, C, D such that,
$\vec{\text{b}}-\vec{\text{a}}=\vec{\text{c}}-\vec{\text{d}}$
Given that,
$\vec{\text{b}}-\vec{\text{a}}=\vec{\text{c}}-\vec{\text{d}}$
$\overrightarrow{\text{AB}}=\overrightarrow{\text{DC}}$
So, AB is parallel and equal to DC (in magnitude).
Hence,
ABCD is a parallelogram.
$\vec{\text{b}}-\vec{\text{a}}=\vec{\text{c}}-\vec{\text{d}}$
Given that,
$\vec{\text{b}}-\vec{\text{a}}=\vec{\text{c}}-\vec{\text{d}}$
$\overrightarrow{\text{AB}}=\overrightarrow{\text{DC}}$
So, AB is parallel and equal to DC (in magnitude).
Hence,
ABCD is a parallelogram.
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