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: Reading off the direction ratios:
The given line has direction ratios \((3,5,6)\) (the denominators).

Step 2: Using the point-direction form:
A line through \((x_1,y_1,z_1)\) with direction ratios \((a,b,c)\) has equation \(\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}\).

Step 3: Substituting the given point and direction:
With \((x_1,y_1,z_1)=(-1,4,-5)\) and \((a,b,c)=(3,5,6)\) (parallel lines share direction ratios).

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