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Question
Write the vector equation of a line given by \[\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2} .\]
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Solution
We have ,
\[\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2} .\]
The given line passes through the point (5, - 4 , 6 ) and has direction ratios proportional to 3, 7, 2.
Vector equation of the given line passing through the point having position vector
\[\overrightarrow{a} = 5 \hat{i} - 4 \hat{j} + 6 \hat{k} \] and parallel to a vector \[\overrightarrow{b} = 3 \hat{i} + 7 \hat{j} + 2 \hat{k} \] is \[\overrightarrow{r} = \overrightarrow{a} + \lambda \overrightarrow{b} \]
\[ \Rightarrow \overrightarrow{r} = 5 \hat{i} - 4 \hat{j} + 6 \hat{k} + \lambda\left( 3 \hat{i} + 7 \hat{j} + 2 \hat{k} \right)\]
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