3. 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).

Answers (1)

We have given points where the line is passing through it;

Consider the line joining the points (4, 7, 8) and (2, 3, 4) is AB and line joining the points  (– 1, – 2, 1) and (1, 2, 5)..is CD.

So, we will find the direction ratios of the lines AB and CD;

Direction ratios of AB are a_{1},b_{1}, c_{1}

(2-4),\ (3-7),\ and\ (4-8)   or  -2,\ -4,\ and\ -4

Direction ratios of CD are a_{2},b_{2}, c_{2}

(1-(-1)),\ (2-(-2)),\ and\ (5-1)   or  2,\ 4,\ and\ 4.

Now, lines AB and CD will be parallel to each other if \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}

Therefore we have now;

\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

\therefore \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}

Hence we can say that AB is parallel to CD.

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