Find the vector equation of the line passing through the point ha… - Vidyadip

Question

Find the vector equation of the line passing through the point having position vector $-\hat{ i }-\hat{ j }+2 \widehat{ k }$ and parallel to the line $\overline{ r }=(\hat{ i }+2 \hat{ j }+3 \widehat{ k })+\mu(3 \hat{ i }+2 \hat{ j }+\widehat{ k }), \mu$ is a parameter

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Answer

Let $\overline{ a }$ be the position vector of the point
$
\therefore \overline{ a }=-\hat{ i }-\hat{ j }+2 \widehat{ k }
$
Equation of given line is $\overline{ r }=(\hat{ i }+2 \hat{ j }+3 \widehat{ k })+\mu(3 \hat{ i }+2 \hat{ j }+\widehat{ k })$
$\therefore$ Direction ratios of the line are $3,2,1$.
Let $\overline{ b }$ be the vector parallel to this line.
$
\therefore \overline{ b }=3 \hat{ i }+2 \hat{ j }+\widehat{ k }
$
The vector equation of a line passing through a point with position vector $\overline{ a }$ and parallel to $\overline{ b }$ is $\overline{ r }=\overline{ a }+\lambda \overline{ b }$.
$\therefore$ Vector equation of the line is $\overline{ r }=(-\hat{ i }-\hat{ j }+2 \widehat{ k })+\lambda(3 \hat{ i }+2 \hat{ j }+\widehat{ k })$

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