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Question
Find the vector equation of a line passing through (2, −1, 1) and parallel to the line whose equations are \[\frac{x - 3}{2} = \frac{y + 1}{7} = \frac{z - 2}{- 3} .\]
Solution
We know that the vector equation of a line passing through a point with position vector `vec a` and parallel to the vector `vec b` is \[\vec{r} = \vec{a} + \lambda \vec{b}\]
Here,
\[\vec{a} = 2 \hat{i} - \hat{j} + \hat{k} \]
\[ \vec{b} = 2 \hat{i} + 7 \hat{j} - 3 \hat{k} \]
Vector equation of the required line is
\[\vec{r} = \left( 2 \hat{i} - \hat{j} + \hat{k} \right) + \lambda \left( 2 \hat{i} + 7 \hat{j} - 3 \hat{k} \right)\]
\[\text{ Here } , \lambda \text{ is a parameter } . \]
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