Question

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

Hint:

Use the point and the parallel vector's components as direction ratios in the standard line formula.

Answer 1

Step 1: Understanding the Concept:
A line through a fixed point, parallel to a given vector, has a standard Cartesian (symmetric) equation.
If the line passes through \((x_1,y_1,z_1)\) and is parallel to a vector with direction ratios \((a,b,c)\), its equation follows a known formula.

Step 2: Key Formula:
\[ \dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c} \]

Step 3: Detailed Explanation:
Here the point is \((5,2,-4)\), so \(x_1=5,y_1=2,z_1=-4\).
The direction ratios come from the parallel vector \(3\hat i+2\hat j-8\hat k\), so \(a=3,b=2,c=-8\).
Substitute these values directly into the standard formula.
\[ \dfrac{x-5}{3}=\dfrac{y-2}{2}=\dfrac{z-(-4)}{-8} \]

Final Answer:
This simplifies to the required Cartesian equation of the line. \[ \boxed{\dfrac{x-5}{3}=\dfrac{y-2}{2}=\dfrac{z+4}{-8}} \]