Question
Give a condition that three vectors $\vec{\text{a}},\ \vec{\text{b}}\text{ and }\vec{\text{c}}$ from the three sides of a triangle. what are the other possibilities?
✓
Answer
Let ABC be a triangle such that $\overrightarrow{\text{BC}}=\vec{\text{a}},\ \overrightarrow{\text{AB}}=\vec{\text{c}}\text{ and }\overrightarrow{\text{CA}}=\vec{\text{b}}$. Then,
$\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}+\overrightarrow{\text{AB}}$
$\Rightarrow\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\overrightarrow{\text{BA}}+\overrightarrow{\text{AB}}$ $\big[\because\ \overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}=\overrightarrow{\text{BA}}\big]$
$\Rightarrow\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\overrightarrow{\text{BB}}$ [Using triangle law]
$\Rightarrow\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\vec0$ [By defination of null vector]
Other possibilities are,
$\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}+\overrightarrow{\text{AB}}$
$\Rightarrow\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\overrightarrow{\text{BA}}+\overrightarrow{\text{AB}}$ $\big[\because\ \overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}=\overrightarrow{\text{BA}}\big]$
$\Rightarrow\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\overrightarrow{\text{BB}}$ [Using triangle law]
$\Rightarrow\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\vec0$ [By defination of null vector]
Other possibilities are,
- $\vec{\text{c}}+\vec{\text{a}}=\vec{\text{b}}$
- $\vec{\text{a}}+\vec{\text{b}}=\vec{\text{c}}$
- $\vec{\text{b}}+\vec{\text{c}}=\vec{\text{a}}$
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