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प्रश्न
By computing the shortest distance, determine whether following lines intersect each other.
`(x - 5)/(4) = (y -7)/(-5) = (z + 3)/(-5) and (x - 8)/(7) = (y - 7)/(1) = (z - 5)/(3)`
उत्तर
The shortest distance between the lines
`(x - 5)/(4) = (y -7)/(-5) = (z + 3)/(-5)` ...(i)
`(x - 8)/(7) = (y - 7)/(1) = (z - 5)/(3)` ...(ii)
∴ from line (i) & (ii)
x1 = 5, y1 = 7, z1 = -3, x2 = 8, y2 = 7, z2 = 5
l1 = 4, m1 = –5, n1 = -5, l2 = 7, m2 = 1, n2 = 3
The given two lines intersect each other if and only if
`|(x_2 - x_1, y_2 - y_1, z_2 - z_1),(l_1, m_1, n_1),(l_2, m_2, n_2)| = 0`
= `|(3, 0, 8),(4, -5, -5),(7, 1, 3)|`
= 3(– 15 + 5) – 0(12 + 35) + 8(4 + 35)
= – 30 – 0 + 312
= 282
`≠` 0
∴ The given lines not intersect each other.
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