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Find the shortest distance between the lines r = (1, 5, 6) + y(-2, -1, 0) and r = (1, 7, -3) + z(2, 0, 4)

Vector joining the two lines = (1, 5, 6) - (1, 7, -3) = (0, -2, 9)Normal vector to the two lines = (-2, -1, 0) x (2, 0, 4) = (-4, 8, 2) = 2(-2, 4, 1)Hence, using the dot product, shortest distance = (0, -2, 9) "dot" (-2, 4, 1) / sqrt(2² + 4² + 1²) = -8 + 9 / sqrt(4 + 16 + 1) = 1/sqrt(21)

Answered by Abhinav H. • Further Mathematics tutor

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Find the shortest distance between the lines r = (1, 5, 6) + y(-2, -1, 0) and r = (1, 7, -3) + z(2, 0, 4)

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The complex number -2sqrt(2) + 2sqrt(6)I can be expressed in the form rexp(iTheta), where r>0 and -pi < theta <= pi. By using the form rexp(iTheta) solve the equation z^5 = -2sqrt(2) + 2sqrt(6)i.