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Question
Find the Cartesian equations of the line passing through A(3, 2, 1) and B(1, 3, 1).
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Solution
The direction ratios of the line AB are 3 – 1, 2 – 3, 1 – 1, i.e., 2, – 1, 0.
The parametric equations of the line passing through (x1, y1, z1) and having direction ratios a, b, c are
x = x1 + aλ, y = y1 + bλ, z = z1 + cλ
∴ the parametric equations of the line passing through (3, 2, 1) and having direction ratios 2, –1, 0 are
x = 3 + 2λ, y = 2 − λ, z = 1 + 0(λ)
∴ x – 3 = 2λ, y − 2 = −λ, z = 1
∴ `(x - 3)/(2) = (y - 2)/(-1)` = λ, z = 1
∴ The Cartesian equations of the required line are:
`(x - 3)/(2) = (y - 2)/(-1)`, z = 1
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