Q.Find the vector equation of the line passing through the point having position vector $-2\hat{i} + \hat{j} + \hat{k}$ and parallel to vector $4\hat{i} - \hat{j} + 2\hat{k}$.

Vector equation of a line through a point $A(\bar a)$ parallel to $\bar b$ is $\bar r=\bar a+\lambda\bar b$.

The line passes through the point with position vector $\bar a = 4\hat i-\hat j+2\hat k$ and is parallel to $\bar b=-2\hat i-\hat j+\hat k$.

The vector equation of a line through a fixed point $\bar a$ and parallel to $\bar b$ is

$\bar r=\bar a+\lambda \bar b$

Substituting,

$\bar r=(4\hat i-\hat j+2\hat k)+\lambda(-2\hat i-\hat j+\hat k)$

This matches option (d) exactly (options (a), (b), (c) alter a sign in either the point or the direction vector).

Vector equation: (point) + $\lambda$(direction).

Line through position vector $\bar a=4\hat i-\hat j+2\hat k$, parallel to $\bar b=-2\hat i-\hat j+\hat k$.

Vector equation: $\bar r=\bar a+\lambda\bar b=(4\hat i-\hat j+2\hat k)+\lambda(-2\hat i-\hat j+\hat k)$, $\lambda\in\mathbb R$

Vector equation of a line through a point $\bar a$ in the direction $\bar b$ is $\bar r = \bar a + \lambda \bar b$.

The line passes through the point with position vector $\bar a = 4\hat i - \hat j + 2\hat k$ and has direction vector $\bar b = -2\hat i + \hat j + \hat k$.

The vector equation of a line through a given point with position vector $\bar a$ along direction $\bar b$ is:

$\bar r = \bar a + \lambda \bar b, \qquad \lambda \in \mathbb{R}$

Substituting:

$\bar r = (4\hat i - \hat j + 2\hat k) + \lambda(-2\hat i + \hat j + \hat k)$