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(A) Let $AB$ be the line passing through the points $A(4,7,8)$ and $B(2,3,4)$.Let $CD$ be the line passing through the points $C(-1,-2,1)$ and $D(1,2,

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Vedclass · Answer

(A) Let $AB$ be the line passing through the points $A(4,7,8)$ and $B(2,3,4)$.
Let $CD$ be the line passing through the points $C(-1,-2,1)$ and $D(1,2,5)$.
The direction ratios $(a_1, b_1, c_1)$ of line $AB$ are given by $(x_2-x_1, y_2-y_1, z_2-z_1)$:
$a_1 = 2-4 = -2$
$b_1 = 3-7 = -4$
$c_1 = 4-8 = -4$
The direction ratios $(a_2, b_2, c_2)$ of line $CD$ are given by $(x_4-x_3, y_4-y_3, z_4-z_3)$:
$a_2 = 1-(-1) = 2$
$b_2 = 2-(-2) = 4$
$c_2 = 5-1 = 4$
Two lines are parallel if their direction ratios are proportional,i.e.,$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.
Calculating the ratios:
$\frac{a_1}{a_2} = \frac{-2}{2} = -1$
$\frac{b_1}{b_2} = \frac{-4}{4} = -1$
$\frac{c_1}{c_2} = \frac{-4}{4} = -1$
Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = -1$,the direction ratios are proportional.
Therefore,the line $AB$ is parallel to the line $CD$.

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

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