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Question
Find the shortest distance between the lines `(x + 1)/(7) = (y + 1)/(-6) = (z + 1)/(1) and (x - 3)/(1) = (y - 5)/(-2) = (z - 7)/(1)`
Solution
The shortest distance between the lines
`(x - x_1)/(l_1) = (y - y_1)/(m_1) = (z - z_1)/(n_1) and (x - x_2)/(l_2) = (y - y_2)/(m_2) = (z - z_2)/(n_2) ` is give n by
d = `||(x_2 - x_1, y_2 - y_1, z_2 - z_1),(l_1, m_1, n_1),(l_2, m_2, n_2)|/sqrt((m_1n_2 - m_2n_1)^2 + (l_2n_1 - 1_1n_2)^2 + (l_1m_2 - l_2m_1)^2)|`
The equation of the given lines are
`(x + 1)/(7) = (y + 1)/(-6) = (z + 1)/(1) and (x - 3)/(1) = (y - 5)/(-2) = (z - 7)/(1)`
∴ x1 = –1, y1 = – 1, z1 = – 1, x2 = 3, y2 = 5, z2 = 7,
l1 = 7, m1 = – 6, n1 = 1, l2 = 1, m2 = – 2, n2 = 1
`|(x_2 - x_1, y_2 - y_1, z_2 - z_1),(l_1, m_1, n_1),(l_2, m_2, n_2)| = |(4, 6, 8),(7, -6, 1),(1, -2, 1)|`
= 4(– 6 + 2) – 6(7 – 1) + 8(– 14 + 6)
= – 16 – 36 – 64
= – 116
and (m1n2 – m2n1)2 + (l2n1 – l1n2)2 + (l1m2 – l2m1)2
= (– 6 + 2)2 + (1 – 7)2 + (– 14 + 6)2
= 16 + 36 + 64
= 116
Hence, the required shortest distance between the given lines
= `|(-116)/sqrt(116)|`
= `sqrt(116)`
= `2sqrt(29) "units"`.
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