Question

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).

Answer 1

Let AB be the line through the points (4,7,8), and (2,3,4), and CD be the line through the points (-1,-2,1), and (1,2,5).

The direction ratios, a₁, b₁, c₁ of AC are (2-4), (3-7), and (4-8) i.e., -2, -4, and -4.
The direction ratios, a₂, b₂, c₂ of CD are (1-(-1)), (2-(-2)), and (5-1) i.e., 2, 4, and 4.

AB will parellel to CD, if \(\frac{a_1}{a_2}\)=\(\frac{b_1}{b_2}\)=\(\frac{c_1}{c_2}\)

\(\frac{a_1}{a_2}\)
=\(-\frac{2}{2}\)
=-1

\(\frac{b_1}{b_2}\)
=\(-\frac{4}{4}\)
=-1

\(\frac{c_1}{c_2}\)
=\(-\frac{4}{4}\)
=-1

∴ \(\frac{a_1}{a_2}\) = \(\frac{b_1}{b_2}\) = \(\frac{c_1}{c_2}\)

Thus, AB is parallel to CD.