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Algebra of Vectors — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Vectors3 Marks
Question
If the vertices A, B, C of a triangle ABC are the points with position vectors $\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}},\ \text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}},\ \text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}$ respectively, what are the vectors determined by its sides? Find the length of these vectors.
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Answer

Given the vertices of a triangle A, B and C with position vectors $\text{a}_1\hat{\text{i}}+\text{a}_2\hat{\text{j}}+\text{a}_3\hat{\text{k}},\ \text{b}_1\hat{\text{i}}+\text{b}_2\hat{\text{j}}+\text{b}_3\hat{\text{k}}$ and $\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}$ respectively. Then,
$\overrightarrow{\text{AB}}=(\text{b}_1-\text{a}_1)\hat{\text{i}}+(\text{b}_2-\text{a}_2)\hat{\text{j}}+(\text{b}_3-\text{a}_3)\hat{\text{k}}$
$\overrightarrow{\text{BC}}=(\text{c}_1-\text{b}_1)\hat{\text{i}}+(\text{c}_2-\text{b}_2)\hat{\text{j}}+(\text{c}_3-\text{b}_3)\hat{\text{k}}$
$\overrightarrow{\text{CA}}=(\text{a}_1-\text{c}_1)\hat{\text{i}}+(\text{a}_2-\text{c}_2)\hat{\text{j}}+(\text{a}_3-\text{c}_3)\hat{\text{k}}$
Therefore, the length of these vectors are:
$\Big|\overrightarrow{\text{AB}}\Big|=\sqrt{(\text{b}_1-\text{a}_1)^2+(\text{b}_2-\text{a}_2)^2+(\text{b}_3-\text{a}_3)^2}$
$\Big|\overrightarrow{\text{BC}}\Big|=\sqrt{(\text{c}_1-\text{b}_1)^2+(\text{c}_2-\text{b}_2)^2+(\text{c}_3-\text{b}_3)^2}$
$\Big|\overrightarrow{\text{CA}}\Big|=\sqrt{(\text{a}_1-\text{c}_1)^2+(\text{a}_2-\text{c}_2)^2+(\text{a}_3-\text{c}_3)^2}$

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