Question

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

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Solution

## $\mathrm{We}\mathrm{know}\mathrm{that}\mathrm{the}\mathrm{direction}\mathrm{ratios}\mathrm{of}\mathrm{the}\mathrm{line}\mathrm{passing}\mathrm{through}\mathrm{the}\mathrm{points}\left({x}_{1},{y}_{1},{z}_{1}\right)\mathrm{and}\left({x}_{2},{y}_{2},{z}_{2}\right)\mathrm{are}{x}_{2}-{x}_{1},{y}_{2}-{y}_{1},{z}_{2}-{z}_{1}.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Let}\mathrm{the}\mathrm{first}\mathrm{two}\mathrm{points}\mathrm{be}\mathrm{A}\left(4,7,8\right)\mathrm{and}\mathrm{B}\left(2,3,4\right).\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Thus},\mathrm{the}\mathrm{dir}\mathrm{ection}\mathrm{ratios}\mathrm{of}\mathrm{AB}\mathrm{are}\left(2-4\right),\left(3-7\right),\left(4-8\right),\mathrm{i}.\mathrm{e}.-2,-4,-4.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Similarly},\mathrm{let}\mathrm{the}\mathrm{other}\mathrm{two}\mathrm{points}\mathrm{be}\mathrm{C}\left(-1,-2,1\right)\mathrm{and}\mathrm{D}\left(1,2,5\right).\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{Thus},\mathrm{the}\mathrm{direction}\mathrm{ratios}\mathrm{of}\mathrm{CD}\mathrm{are}\left[1-\left(-1\right)\right],\left[2-\left(-2\right)\right],\left(5-1\right),\mathrm{i}.\mathrm{e}.2,4,4.\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{It}\mathrm{can}\mathrm{be}\mathrm{seen}\mathrm{that}\mathrm{the}\mathrm{direction}\mathrm{ratios}\mathrm{of}\mathrm{CD}\mathrm{are}-1\mathrm{times}\mathrm{that}\mathrm{of}\mathrm{AB},\mathrm{i}.\mathrm{e}.\mathrm{they}\mathrm{are}\mathrm{proportional}.\phantom{\rule{0ex}{0ex}}\mathrm{Therefore},\mathrm{AB}\mathrm{and}\mathrm{CD}\mathrm{are}\mathrm{parallel}\mathrm{lines}.$

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