Question

The vector equation of the line passing through $\text{P}(1,2,3)$ and $\text{Q}(2,3,4)$ is

• A. $(\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$
• B. $(\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} - \hat{k})$
• C. $(\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$
• D. $(\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 6\hat{j} + 12\hat{k})$

Hint:

To quickly verify a two-point line equation, calculate the direction ratios by subtracting the coordinates directly: $2-1=1$, $3-2=1$, $4-3=1$. The direction ratios are $(1, 1, 1)$, meaning the parameter $\lambda$ must be multiplied by $(\hat{i} + \hat{j} + \hat{k})$. This instantly singles out option (A).

Answer 1

The Correct Option is A

Step 1: Recall the form of a line through two points.
A line through points with position vectors $\vec{a}$ and $\vec{b}$ is $\vec{r} = \vec{a} + \lambda(\vec{b} - \vec{a})$. The base point is one of the given points; the direction is the difference.
Step 2: Write the position vector of P.
$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, since $P = (1, 2, 3)$.
Step 3: Write the position vector of Q.
$\vec{b} = 2\hat{i} + 3\hat{j} + 4\hat{k}$, since $Q = (2, 3, 4)$.
Step 4: Find the direction vector.
$\vec{b} - \vec{a} = (2-1)\hat{i} + (3-2)\hat{j} + (4-3)\hat{k} = \hat{i} + \hat{j} + \hat{k}$.
Step 5: Assemble the equation.
$\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$.
Step 6: Confirm both points lie on it.
$\lambda = 0$ gives P, and $\lambda = 1$ gives $2\hat{i} + 3\hat{j} + 4\hat{k} = Q$. The line is correct.
\[ \boxed{\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})} \]