Vector-Algebra Question 414

Question: If $ \overset{\to }{\mathop{a}},,\overset{\to }{\mathop{b}},,\overset{\to }{\mathop{c}}, $ are the position vectors of corners A, B, C of a parallelogram ABCD, then what is the position vector of the corner D?

Options:

A) $ \overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},+\overset{\to }{\mathop{c}}, $

B) $ \overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},-\overset{\to }{\mathop{c}}, $

C) $ \overset{\to }{\mathop{a}},-\overset{\to }{\mathop{b}},+\overset{\to }{\mathop{c}}, $

D) $ -\overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},+\overset{\to }{\mathop{c}}, $

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Answer:

Correct Answer: C

Solution:

  • [c] Let O be the origin and ABCD be the parallelogram. In $ \Delta ,ODC, $ $ \overrightarrow{OD}=\overrightarrow{OC}+\overrightarrow{CD} $ $ \overrightarrow{CD}=-\overrightarrow{AB} $ and $ \overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA} $ [In $ \Delta ,AOB $ ] $ =\overset{\to }{\mathop{b}},-\overset{\to }{\mathop{a}}, $ Thus, $ \overrightarrow{CD}=-\overrightarrow{AB}=\overrightarrow{a}-\overrightarrow{b} $ So, $ \overrightarrow{OD}=\overrightarrow{c}+\overrightarrow{a}-\overrightarrow{b} $ [since, $ \overrightarrow{OC}=\overrightarrow{C} $ and $ \overrightarrow{CD}=\overrightarrow{a}-\overrightarrow{b} $ ]