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Question
Find the vector equation of the line passing through the point having position vector `hat"i" + 2hat"j" + 3hat"k" "and perpendicular to vectors" hat"i" + hat"j" + hat"k" and 2hat"i" - hat"j" + hat"k"`.
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Solution
Let `bar"b" = hat"i" + hat"j" + hat"k" and bar"c" = 2hat"i" - hat"j" + hat"k"`
The vector perpendicular to the vectors `bar"b" and bar"c"` is given by
`bar"b" xx bar"c" = |(hat"i",hat"j",hat"k"),(1,1,1),(2,-1,1)|`
= `hat"i"(1 + 1) - hat"j"(1 - 2) + hat"k"(-1 - 2)`
= `2hat"i" + hat"j" - 3hat"k"`
Since the line is perpendicular to the vector `bar"b" and bar"c", "it is parallel to" bar"b" xx bar"c"`.
The vector equation of the line passing through `"A"(bara) "and parallel to" bar"b" xx bar"c" "is" bar"r" = bar"a" + λ(bar"b" xx bar"c")`, where λ is a scalar.
Here, `bar"a" = hat"i" + 2hat"j" + 3hat"k"`
Hence, the vector equation of the required line is `bar"r" = (hat"i" + 2hat"j" + 3hat"k") + λ(2hat"i" + hat"j" - 3hat"k")`.
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