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प्रश्न
Prove that the points having position vectors \[\hat{i} + 2 \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k} , - 3 \hat{i} - 2 \hat{i} - 5 \hat{k}\] are collinear.
उत्तर
Let A, B, C be the points with position vectors
\[\hat{i} + 2 \hat{j} + 3 \hat{k} , 3 \hat{i} + 4 \hat{j} + 7 \hat{k} , - 3 \hat{i} - 2 \hat{j} - 5 \hat{k} .\]
Then,
\[\overrightarrow{AB} =\] Position vector of B - Position vector of A
\[= 3 \hat{i} + 4 \hat{j} + 7 \hat{k} - \hat{i} - 2 \hat{j} - 3 \hat{k} \]
\[ = 2 \hat{i} + 2 \hat{j} + 4 \hat{k}\]
\[\overrightarrow{BC} =\] Position vector of C - Position vector of B
\[= - 3 \hat{i} - 2 \hat{j} - 5 \hat{k} - 3 \hat{i} - 4 \hat{j} - 7 \hat{k} \]
\[ = - 6 \hat{i} - 6 \hat{j} - 12 \hat{k} \]
\[ = - 3\left( 2 \hat{i} + 2 \hat{j} + 4 \hat{k} \right)\]
∴ \[\overrightarrow{BC} = - 3 \overrightarrow{AB}\]
\[\text{ So, }\overrightarrow{AB} \text{ and }\overrightarrow{BC}\] are parallel vectors.
But B is a point common to them.
So,
\[\overrightarrow{AB}\] and \[\overrightarrow{BC}\] are collinear.
Hence,
A, B, C are collinear.
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