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Question
Find the vector equation of the line passing through the point having position vector `-hat"i"- hat"j" + 2hat"k"` and parallel to the line `bar"r" = (hat"i" + 2hat"j" + 3hat"k") + mu(3hat"i" + 2hat"j" + hat"k")`, µ is a parameter
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Solution
Let `bar"a"` be the position vector of the point
∴ `bar"a" = -hat"i" - hat"j" + 2hat"k"`
Equation of given line is `bar"r" = (hat"i" + 2hat"j" + 3hat"k") + mu(3hat"i" + 2hat"j" + hat"k")`
∴ Direction ratios of the line are 3, 2, 1.
Let `bar"b"` be the vector parallel to this line.
∴ `bar"b" = 3hat"i" + 2hat"j" + hat"k"`
The vector equation of a line passing through a point with position vector `bar"a"` and parallel to `bar"b"` is `bar"r" = bar"a" + lambdabar"b"`.
∴ Vector equation of the line is `bar"r" = (-hat"i" - hat"j" + 2hat"k") + lambda(3hat"i" + 2hat"j" + hat"k")`
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