Find the Cartesian equation of the line passing through the point $(5, 2, - 4)$ and parallel to the vector $3^i + 2^j - 8^k$ .

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Solution

The Cartesian equation of the line passing through the point $(x_{1}, y_{1}, z_{1})$ and parallel to the vector $a^i + b^j + c^k$ is $\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}$

Let $x_{1} = 5, y_{1} = 2, z_{1} = - 4$ and $a = 3, b = 2, c = - 8$

Hence the equation of the line in cartesian form is $\frac{x - 5}{3} = \frac{y - 2}{2} = \frac{z - (- 4)}{- 8}$

$\Rightarrow \frac{x - 5}{3} = \frac{y - 2}{2} = \frac{z + 4}{- 8}$

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