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: Understanding the Question:
The question asks for the vector equation of a line passing through two specified coordinate points, P and Q, in three-dimensional space.

Step 2: Key Formula or Approach:
The vector equation of a straight line passing through a point with position vector $\vec{a}$ and moving along a direction vector $\vec{m}$ is written as: $$\vec{r} = \vec{a} + \lambda \vec{m}$$ When the line passes through two points $\text{P}(\vec{a})$ and $\text{Q}(\vec{b})$, the direction vector is simply the difference vector between them: $$\vec{m} = \vec{b} - \vec{a}$$

Step 3: Detailed Explanation:
1. Identify the position vector of the first passing point $\text{P}(1, 2, 3)$, which serves as our base vector $\vec{a}$: $$\vec{a} = 1\hat{i} + 2\hat{j} + 3\hat{k} = \hat{i} + 2\hat{j} + 3\hat{k}$$ 2. Identify the position vector of the second point $\text{Q}(2, 3, 4)$, denoted as $\vec{b}$: $$\vec{b} = 2\hat{i} + 3\hat{j} + 4\hat{k}$$ 3. Compute the direction vector $\vec{m}$ by taking the difference $\vec{b} - \vec{a}$: $$\vec{m} = (2 - 1)\hat{i} + (3 - 2)\hat{j} + (4 - 3)\hat{k}$$ $$\vec{m} = 1\hat{i} + 1\hat{j} + 1\hat{k} = \hat{i} + \hat{j} + \hat{k}$$ 4. Substitute $\vec{a}$ and $\vec{m}$ into the standard line formulation: $$\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$$

Step 4: Final Answer:
The correct vector equation is $(\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} + \hat{j} + \hat{k})$, which matches option (A).