Find the equation of a line passing through the point (1, 2, -4) and perpendicular to two lines
$\begin{matrix} r^{} = (8 i - 19 j + 10 k) + λ (3 i - 16 j + 7 k) and r^{} = (15 i + 29 j + 5 k) + μ (3 i + 8 j - 5 k) . \end{matrix}$

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Solution

Equation of line three $(1, 2, - 4)$
$\frac{x - 1}{a} = \frac{y - 2}{b} = \frac{z + 4}{c}$
$L i n e i s ∥ t o : (3 ˆ i - 16 ˆ j + 7 ˆ k) \times (3 ˆ i + 8 ˆ j - 5 ˆ k)$
$= 24 ˆ i + 36 ˆ j + 72 ˆ k$
$= (2 ˆ i + 3 ˆ j + 6 ˆ k)$
Hence, the equation of line is $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z + 4}{6}$ .

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\vec{r} = (\hat{i} - 2 \hat{j} + \hat{k}) + λ (2 \hat{i} + 3 \hat{j} - 5 \hat{k}) .

\vec{r} \cdot (\hat{i} + 2 \hat{j} - 5 \hat{k}) + 9 = 0 .

\vec{r} \cdot (\hat{i} + 2 \hat{j} - 5 \hat{k}) + 9 = 0 .

2 ˆ i + 3 ˆ j - 7 ˆ k

r = (ˆ i - 2 ˆ j - ˆ k) + λ (3 ˆ i - 8 ˆ j + 2 ˆ k)

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