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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") + λ(3hat"i" + 2hat"j" + hat"k").`
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Solution
Let A be point having position vector `bar"a" = -hat"i" - hat"j" + 2hat"k"`
The required line is parallel to the line
`bar"r" = (hat"i" + 2hat"j" + 3hat"k" + λ(3hat"i" + 2hat"j" + hat"k")`
∴ it is parallel to the vector
`bar"b" = 3hat"i" + 2hat"j" + hat"k"`
The vector equation of the line passing through `"A"(bara) "and parallel to" bar"b" "is" bar"r" = bar"a" + λbar"b"` where λ is a scalar.
∴ the required vector equation of the line is
`bar"r" = (-hat"i" - hat"j" + 2hat"k") + λ(3hat"i" + 2hat"j" + hat"k")`.
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