Find the equations of the line through $(3, 5, 4)$ and parallel to the line $x + 2 y - 2 z - 7 = 0 = 6 x + 8 y - 9 z - 1$ .

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Solution

We have,
$\begin{matrix} π_{1} : x + 2 y - 2 z = 7 π_{2} : 6 x + 8 y - 9 z = 1 \end{matrix}$
Line of intersection of $π_{1}$ and $π_{2}$ is parallel to $π_{1} \times π_{2}$ .
$∣ ∣ ∣ ∣ ∣ \begin{matrix} ˆ i & ˆ j & ˆ k 1 & 2 & - 2 6 & 8 & - 9 \end{matrix} ∣ ∣ ∣ ∣ ∣ = ˆ i (- 18 + 16) + ˆ j (- 9 + 12) + ˆ k (8 - 12)$
$- 2 ˆ i - 3 ˆ j - 4 ˆ k$
Equation of line $\frac{x - 3}{- 2} = \frac{y - 5}{- 3} = \frac{z - 4}{- 4}$

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