Question
Show that the points $2\hat{\text{i}},-\hat{\text{i}}-4\hat{\text{j}}\text{ and }-\hat{\text{i}}+4\hat{\text{j}}$ form an isosceles triangle.
✓
Answer
Given:- The points A, B, C with position vectors $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ respectively.
Also, $\vec{\text{a}}=2\hat{\text{i}}$
$\vec{\text{b}}=-\hat{\text{i}}-4\hat{\text{j}}$
$\vec{\text{c}}=-\hat{\text{i}}+4\hat{\text{j}}$
Then,
$\overrightarrow{\text{AB}}=\vec{\text{b}}-\vec{\text{a}}$
$\Rightarrow\overrightarrow{\text{AB}}=\big(-\hat{\text{i}}-4\hat{\text{j}}\big)-2\hat{\text{i}}$
$\Rightarrow\overrightarrow{\text{AB}}=-3\hat{\text{i}}-4\hat{\text{j}}$
Now, $\Big|\overrightarrow{\text{AB}}\Big|=\sqrt{(-3)^2+(-4)^2}$
$=\sqrt{9+16}$
$=\sqrt{25}$
$=5$
$\overrightarrow{\text{BC}}=\vec{\text{c}}-\vec{\text{b}}$
$\Rightarrow\overrightarrow{\text{BC}}=\big(-\hat{\text{i}}+4\hat{\text{j}}\big)-\big(-\hat{\text{i}}-4\hat{\text{j}}\big)$
$\Rightarrow\overrightarrow{\text{BC}}=-\hat{\text{i}}+4\hat{\text{j}}+\hat{\text{i}}+4\hat{\text{j}}$
$\Rightarrow\overrightarrow{\text{BC}}=8\hat{\text{j}}$
Also, $\vec{\text{a}}=2\hat{\text{i}}$
$\vec{\text{b}}=-\hat{\text{i}}-4\hat{\text{j}}$
$\vec{\text{c}}=-\hat{\text{i}}+4\hat{\text{j}}$
Then,
$\overrightarrow{\text{AB}}=\vec{\text{b}}-\vec{\text{a}}$
$\Rightarrow\overrightarrow{\text{AB}}=\big(-\hat{\text{i}}-4\hat{\text{j}}\big)-2\hat{\text{i}}$
$\Rightarrow\overrightarrow{\text{AB}}=-3\hat{\text{i}}-4\hat{\text{j}}$
Now, $\Big|\overrightarrow{\text{AB}}\Big|=\sqrt{(-3)^2+(-4)^2}$
$=\sqrt{9+16}$
$=\sqrt{25}$
$=5$
$\overrightarrow{\text{BC}}=\vec{\text{c}}-\vec{\text{b}}$
$\Rightarrow\overrightarrow{\text{BC}}=\big(-\hat{\text{i}}+4\hat{\text{j}}\big)-\big(-\hat{\text{i}}-4\hat{\text{j}}\big)$
$\Rightarrow\overrightarrow{\text{BC}}=-\hat{\text{i}}+4\hat{\text{j}}+\hat{\text{i}}+4\hat{\text{j}}$
$\Rightarrow\overrightarrow{\text{BC}}=8\hat{\text{j}}$
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