Q.State whether the following statement is True or False: The equation of a line, which is parallel to $2\hat{i} + 3\hat{j} + \hat{k}$ and which passes through the point $(5, -2, 4)$, is $\dfrac{x-5}{2} = \dfrac{y+2}{-1} = \dfrac{z-4}{3}$.

Concept: Vector Equation Of Line — A line parallel to a given vector $\vec{b}$ and passing through a point with position vector $\vec{a}$ has equation $\vec{r} = \vec{a} + \lambda \vec{b}$. In Cartesian form, if $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$, the line is $\frac{x - x_1}{b_1} = \frac{y - y_1}{b_2} = \frac{z - z_1}{b_3}$.

Step 1: The given line is parallel to $2\hat{i} + 3\hat{j} + \hat{k}$, so its direction ratios are $(2, 3, 1)$.

Step 2: The line passes through $(5, -2, 4)$. Using the Cartesian form, the correct equation is: …

The core idea here is simple: a line's direction is determined by its direction ratios (or direction vector). If two lines are parallel, their direction vectors must be scalar multiples of each other. The given equation claims a specific direction, so we just check whether that direction matches the required one.

Let’s break it down.

1. What does "parallel to a vector" mean for a line?

   If a line is parallel to a vector $\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}$, then the direction ratios of the line are $(a, b, c)$ — or any scalar multiple of them. So the line we want must have direction ratios proportional to $(2, 3, 1)$.

2. What direction does the given equation represent?

   The symmetric form of a line is

where $(a, b, c)$ are the direction ratios. Here, the equation is

So the direction ratios claimed are $(2, -1, 3)$.

3. Are $(2, -1, 3)$ proportional to $(2, 3, 1)$? For two sets of numbers to be proportional, the ratios of corresponding components must be equal. Check:

Clearly, $1 \neq -\frac{1}{3} \neq 3$. They are not proportional. …

Method: Line Through a Point Parallel to a Given Vector

When a line must be parallel to a vector $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ and pass through $(x_1,y_1,z_1)$, its direction ratios are exactly the components of $\vec{b}$.

Steps

Step 1: Take the direction ratios straight from the parallel vector.

"Parallel to $\vec{b}$" means the direction ratios are $(b_1,b_2,b_3)$ — or any non-zero multiple. The point never changes the direction.

Step 2: Place the point in the numerators, the direction ratios in the denominators.

$\frac{x-x_1}{b_1}=\frac{y-y_1}{b_2}=\frac{z-z_1}{b_3}.$ …

Common Mistakes

Mistake 1: Checking only the first direction ratio.

Why it's wrong: $(2,-1,3)$ and $(2,3,1)$ agree in the first slot but differ afterwards, so they are not parallel; one matching ratio proves nothing. Correct approach: confirm all three ratios are in the same proportion before concluding parallelism.

Mistake 2: Thinking a correct point makes the equation correct. …