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Question
Find Cartesian equation of the line passing through the point A(2, 1, −3) and perpendicular to vectors `hat"i" + hat"j" + hat"k"` and `hat"i" + 2hat"j" - hat"k"`
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Solution
Let `bar"b" = hat"i" + hat"j" + hat"k"` and `bar"c" = hat"i" + 2hat"j" - hat"k"`
We know that `bar"b" xx bar"c"` is perpendicular to both `bar"b"` and `bar"c"`.
∴ `bar"b" xx bar"c" = |(hat"i", hat"j", hat"k"),(1, 1, 1),(1, 2, -1)|`
= `hat"i"(-1, -2) -hat"j"(-1, -1) + hat"k"(2 - 1)`
= `-3hat"i" + 2hat"j" + hat"k"`
∴ The direction ratios of the required line are – 3, 2, 1 and it passes through A(2, 1, – 3).
∴ The Cartesian equation of a line passing through the point (x1, y1, z1) and having direction ratios (a, b, c) is
`(x - x_1)/"a" = (y - y_1)/"b" = (z - z_1)/"c"`
i.e., `(x - 2)/(-3) = (y - 1)/2 = (z + 3)/1`
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