Question

The equation of the line passing through (0, 0, 1) and (1, 1, 0) is ________.

• A. $\vec{r}=\hat{k}+\lambda(\hat{i}+\hat{j}-\hat{k}),\lambda\in\mathbb{R}$
• B. $\vec{r}=\hat{j}+\lambda(\hat{i}-\hat{j}+\hat{k}),\lambda\in\mathbb{R}$
• C. $\vec{r}=\hat{i}+\lambda(\hat{i}+\hat{j}+\hat{k}),\lambda\in\mathbb{R}$
• D. $\vec{r}=\hat{i}+\hat{j}+\lambda(\hat{i}-\hat{j}-\hat{k}),\lambda\in\mathbb{R}$
• E. $\vec{r}=\hat{i}+\hat{j}+\hat{k}+\lambda(\hat{i}+\hat{j}-\hat{k}),\lambda\in\mathbb{R}$

Hint:

The direction vector is found by subtracting the coordinates of the two points.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The vector equation of a line passing through a point with position vector \(\vec{a}\) and parallel to a vector \(\vec{b}\) is given by \(\vec{r} = \vec{a} + \lambda\vec{b}\), where \(\lambda\) is a scalar parameter. To find the equation of a line passing through two points, we can use one point as the base point (\(\vec{a}\)) and the vector connecting the two points as the direction vector (\(\vec{b}\)).
Step 2: Key Formula or Approach
Let the two given points be P and Q with position vectors \(\vec{p}\) and \(\vec{q}\).
1. The position vector \(\vec{a}\) can be the position vector of either point, e.g., \(\vec{a} = \vec{p}\).
2. The direction vector \(\vec{b}\) is the vector from P to Q, which is \(\vec{b} = \vec{q} - \vec{p}\).
The equation of the line is then \(\vec{r} = \vec{p} + \lambda(\vec{q} - \vec{p})\).
Step 3: Detailed Explanation
1. Identify the position vectors of the given points.
Let point P be (0, 0, 1). Its position vector is \(\vec{p} = 0\hat{i} + 0\hat{j} + 1\hat{k} = \hat{k}\).
Let point Q be (1, 1, 0). Its position vector is \(\vec{q} = 1\hat{i} + 1\hat{j} + 0\hat{k} = \hat{i} + \hat{j}\).
2. Choose a base point.
We can choose either P or Q. Let's choose P as the base point, so \(\vec{a} = \vec{p} = \hat{k}\).
3. Calculate the direction vector.
The direction vector \(\vec{b}\) is the vector from P to Q.
\[ \vec{b} = \vec{q} - \vec{p} = (\hat{i} + \hat{j}) - (\hat{k}) = \hat{i} + \hat{j} - \hat{k} \] 4. Write the equation of the line.
Using the formula \(\vec{r} = \vec{a} + \lambda\vec{b}\):
\[ \vec{r} = \hat{k} + \lambda(\hat{i} + \hat{j} - \hat{k}) \] This matches option (A).
Alternative using point Q as base:
If we had chosen Q as the base point, \(\vec{a} = \vec{q} = \hat{i} + \hat{j}\). The direction vector could be \(\vec{p} - \vec{q} = -(\hat{i} + \hat{j} - \hat{k})\). The equation would be \(\vec{r} = (\hat{i} + \hat{j}) + \mu(-\hat{i} - \hat{j} + \hat{k})\). This is an equivalent representation but doesn't match the options directly. The form in option (A) is the standard one derived. Step 4: Final Answer
The equation of the line is \(\vec{r} = \hat{k} + \lambda(\hat{i} + \hat{j} - \hat{k}), \lambda \in \mathbb{R}\).