Question

Find the Cartesian equation of the line which passes through the point \((-1,4,-5)\) and is parallel to \(\dfrac{x+3}{3}=\dfrac{y-4}{5}=\dfrac{z-8}{6}\).

Hint:

Parallel lines share direction ratios; use the point-direction Cartesian form.

Answer 1

Step 1: Vector form first:
Position vector of the given point: \(\vec r_0=-\hat i+4\hat j-5\hat k\); direction vector (parallel to the given line): \(\vec d=3\hat i+5\hat j+6\hat k\).

Step 2: Writing the vector equation:
\(\vec r=\vec r_0+\lambda\vec d=(-1+3\lambda)\hat i+(4+5\lambda)\hat j+(-5+6\lambda)\hat k\).

Step 3: Converting to Cartesian by eliminating λ:
From each component: \(\lambda=\dfrac{x+1}{3}=\dfrac{y-4}{5}=\dfrac{z+5}{6}\).

Final Answer:
\[ \boxed{\dfrac{x+1}{3}=\dfrac{y-4}{5}=\dfrac{z+5}{6}} \]