Question
If $\vec{\text{a}}\text{ and }\vec{\text{b}}$ are two non-collinear vectors having the same initial point. What are the vectors represented by $\vec{\text{a}}+\vec{\text{b}}\text{ and }\vec{\text{a}}-\vec{\text{b}}$.
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Answer
Here, it is given that $\vec{\text{a}}\text{ and }\vec{\text{b}}$ are two non-collinear vectors having the same initial point.Let $\vec{\text{a}}=\overrightarrow{\text{AB}}\text{ and }\vec{\text{b}}=\overrightarrow{\text{AD}}$, So we can draw a parallelogram ABCD as above.
By the properties of parallelogram
$\overrightarrow{\text{BC}}=\vec{\text{b}}\text{ and }\overrightarrow{\text{DC}}=\vec{\text{a}}$
In $\triangle\text{ABC}$,
Using triangle law,
$\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}=\overrightarrow{\text{AC}}$
$\vec{\text{a}}+\vec{\text{b}}=\overrightarrow{\text{AC}}\ \dots(\text{i})$
In $\triangle\text{ABD}$
$\overrightarrow{\text{AD}}+\overrightarrow{\text{DB}}=\overrightarrow{\text{AB}}$
$\vec{\text{b}}+\overrightarrow{\text{DB}}=\vec{\text{a}}$
$\overrightarrow{\text{DB}}=\vec{\text{a}}-\vec{\text{b}}\ \dots(\text{ii})$
From (i) and (ii), we get that
$\vec{\text{a}}+\vec{\text{b}}\text{ and }\vec{\text{a}}-\vec{\text{b}}$ are diagonals of a parallelogram whose adjacent sides are $\vec{\text{a}}\text{ and }\vec{\text{b}}$
By the properties of parallelogram
$\overrightarrow{\text{BC}}=\vec{\text{b}}\text{ and }\overrightarrow{\text{DC}}=\vec{\text{a}}$
In $\triangle\text{ABC}$,
Using triangle law,
$\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}=\overrightarrow{\text{AC}}$
$\vec{\text{a}}+\vec{\text{b}}=\overrightarrow{\text{AC}}\ \dots(\text{i})$
In $\triangle\text{ABD}$
$\overrightarrow{\text{AD}}+\overrightarrow{\text{DB}}=\overrightarrow{\text{AB}}$
$\vec{\text{b}}+\overrightarrow{\text{DB}}=\vec{\text{a}}$
$\overrightarrow{\text{DB}}=\vec{\text{a}}-\vec{\text{b}}\ \dots(\text{ii})$
From (i) and (ii), we get that
$\vec{\text{a}}+\vec{\text{b}}\text{ and }\vec{\text{a}}-\vec{\text{b}}$ are diagonals of a parallelogram whose adjacent sides are $\vec{\text{a}}\text{ and }\vec{\text{b}}$
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