Find the point where line which passes through point (1, 2, 3) and is parallel to vector line $r = i - j + 2 k + λ (i - 2 j + 3 k)$ meets the xy-plane.

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Solution

Equation of line passing through point $(1, 2, 3)$ and prallel to vector line $^i -^j + 2^k + λ (^i - 2^j + 3^k)$
is $^i + 2^j + 3^k + λ (^i - 2^j + 3^k)$
or $\frac{x - 1}{1} = \frac{y - 2}{- 2} = \frac{z - 3}{3} = λ$
when the line meets $x y$ plane the $z$ co-ordinate is $0$ then $\frac{0 - 3}{3} = λ$
$λ = - 1 ∴ \frac{x - 1}{1} = - 1 \Rightarrow x = 1 - 1 = 0$
and $\frac{y - 2}{- 2} = - 1 \Rightarrow y - 2 = 2 y = 4$
$∴$ co-ordinate ofthe required point is $(0, 4, 0)$

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