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Question
If the vertices of a triangle are the points with position vectors \[a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} , b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} , c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k} ,\]
what are the vectors determined by its sides? Find the length of these vectors.
Solution
Given the vertices of a triangle A, B and C with position vectors
\[a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} , b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}\] and \[c_1 \hat{i} + c_2 \hat{j} + c_3 \hat{k} ,\] respectively. Then,
\[\overrightarrow{AB} = ( b_1 - a_1 ) \hat{i} + ( b_2 - a_2 ) \hat{j} + ( b_3 - a_3 ) \hat{k} . \]
\[ \overrightarrow{BC} = ( c_1 - b_1 ) \hat{i} + ( c_2 - b_2 ) \hat{j} + ( b_3 - a_3 ) \hat{k} . \]
\[ \overrightarrow{CA} = ( a_1 - c_1 ) \hat{i} + ( a_2 - c_2 ) \hat{j} + ( a_3 - c_3 ) \hat{k} .\]
Therefore, the length of these vectors are:
\[\left| \overrightarrow{AB} \right| = \sqrt{( b_1 - a_1 )^2 + ( b_2 - a_2 )^2 + ( b_3 - a_3 )^2} . \]
\[\left| \overrightarrow{BC} \right| = \sqrt{( c_1 - b_1 )^2 + ( c_2 - b_2 )^2 + ( c_3 - b_3 )^2} . \]
\[\left| \overrightarrow{CA} \right| = \sqrt{( a_1 - c_1 )^2 + ( a_2 - c_2 )^2 + ( a_3 - c_3 )^2} .\]
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