Q.Find the cartesian equation of the line which passes through the point $(-2, 4, -5)$ and parallel to the line given by $\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}$.

Concept: Direction Vectors — A line parallel to another line shares the same direction vector. The given line’s symmetric form shows its direction ratios directly.

Step 1: From $\frac{x+3}{3} = \frac{y-4}{5} = \frac{z+8}{6}$, the direction ratios are $(3, 5, 6)$.

Step 2: The required line passes through $(-2, 4, -5)$ and has the same direction vector $(3, 5, 6)$.

Step 3: Using the standard symmetric form $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$, substitute: …

The key idea here is that parallel lines have the same direction. In 3D geometry, the direction of a line is given by its direction vector — the denominators in the symmetric (cartesian) form. Once you know the direction vector and a point on the line, you can write the equation directly.

Let’s unpack the given line first. The equation

is in symmetric form. This means the line passes through $(-3, 4, -8)$ and has direction vector $\vec{d} = (3, 5, 6)$. The denominators are the components of the direction vector.

Now, any line parallel to this one must have the same direction vector $(3,5,6)$. The only thing that changes is the point it passes through. Here, that point is $(-2, 4, -5)$.

So we write the symmetric equation for a line through $(x_1, y_1, z_1)$ with direction $(a,b,c)$ as

Substitute $(-2,4,-5)$ and $(3,5,6)$:

1. For $x$: $x - (-2) = x + 2$, denominator $3$.
2. For $y$: $y - 4$, denominator $5$.
3. For $z$: $z - (-5) = z + 5$, denominator $6$.

Thus the equation is

$\frac{x+2}{3} = \frac{y-4}{5} = \frac{z+5}{6}.$ …

Method: Equation of a line parallel to a given line

A line parallel to a known line borrows that line's direction but keeps its own point. Only the numerators (the point) change; the denominators (the direction) are copied unchanged.

Steps

Step 1: Extract the direction of the given line. In $\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}$ the denominators $(a,b,c)$ are the direction ratios — that is all you take from the given line.

Step 2: Use the new point. The required line passes through its own point $(x_0,y_0,z_0)$; ignore the given line's point entirely. …

Common Mistakes

Mistake 1: Copying the given line's numerators into the answer.

Why it's wrong: the new line passes through $(-2,4,-5)$, not the given line's point $(-3,4,-8)$ — only the direction is shared. Correct approach: use the new point's coordinates: $\frac{x+2}{3}=\frac{y-4}{5}=\frac{z+5}{6}$.

Mistake 2: A sign error turning the point into the numerator.

Why it's wrong: $x-(-2)=x+2$ and $z-(-5)=z+5$; a sign slip shifts the line. Correct approach: substitute carefully into $x-x_0$, $y-y_0$, $z-z_0$. …