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Question
Find the Cartesian equations of the line which passes through the point (2, 1, 3) and perpendicular to the lines `(x - 1)/(1) = (y - 2)/(2) = (z - 3)/(3) and x/(-3) = y/(2) = z/(5)`.
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Solution
Let line L1 be `(x - 1)/(1) = (y - 2)/(2) = (z - 3)/(3)`
Let line L2 be `x/(-3) = y/(2) = z/(5)`
It is perpendicular to the vector `bar"b" = hat"i" + 2hat"j" + hat"k" and bar"c" = -3hat"i" + 2hat"j" + 5hat"k"`.
∴ It is perpendicular to lines whose direction ratios are 1, 2, 1 and -3, 2, 5.
Now, `bar"b"xxbar"c"=|(hat"i",hat"j",hat"k"),(1,2,3),(-3,2,5)|`
`=hat"i"(10-6)-hat"j"(5+9)+hat"k"(2+6)`
`bar"b"xxbar"c"= 4hat"i"-14hat"j"+8hat"k"`
∴ Is parallel to the required line.
The direction ratios of the parallel line are 4, -14, 8 or 2, -7, 4
Let the required line passing through point A = (2, 1, 3) = (x1 y1 z1)
∴ The cartesian equations of the line is
`(x - x_1)/a = (y - y_1)/b = (z - z_1)/c`
`(x - 2)/(2) = (y - 1)/(-7) = (z - 2)/(4)`
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