Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (–1, –2, 1), (1, 2, 5).

We know that

Two lines with direction ratios a_{1}, b_{1}, c_{1} and a_{2}, b_{2}, c_{2} are parallel if the angle between them is θ = 0°, i. e.

⇒

Also, we know that the direction ratios of the line segment joining (x_{1}, y_{1}, z_{1}) and (x_{2}, y_{2}, z_{2}) is taken as x_{2} – x_{1}, y_{2} – y_{1}, z_{2} – z_{1} (or x_{1} – x_{2}, y_{1} – y_{2}, z_{1} – z_{2}).

⇒ The direction ratios of the line through the points (4, 7, 8) and (2, 3, 4) is:

a_{1} = 2 – 4 = -2, b_{1} = 3 – 7 = -4, c_{1} = 4 – 8 = -4

And the direction ratios of the line through the points (– 1, – 2, 1) and (1, 2, 5) is:

a_{2} = 1 – (-1) = 1 + 1 = 2, b_{2} = 2 – (-2) = 2 + 2 = 4, c_{2} = 5 – 1 = 4

Consider

⇒

∴ The line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (–1, –2, 1), (1, 2, 5).

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