If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (–4, 3, –6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.

Angle between the lines with direction ratios a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} is given by

A line passing through A(x_{1}, y_{1}, z_{1}) and B(x_{2}, y_{2}, z_{2}) has direction ratios (x_{1} - x_{2}), (y_{1} - y_{2}), (z_{1} - z_{2})

Direction ratios of line joining the points A(1,2,3) and B(4,5,7)

= (4 - 1), (5 - 2), (7 - 3)

= (3,3,4)

∴ a_{1} = 3, b_{1} = 3, c_{1} = 4

Direction ratios of line joining the points C(-4, 3, -6) and B(2,9,2)

= (2 - (-4)), (9 - 3), (2-(-6))

= (6,6,8)

∴ a_{2} = 6, b_{2} = 6, c_{2} = 8

Now,

∴ cosθ = 1

So, θ = 0°

Hence, Angle between the lines AB and CD is 0°.

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