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: Concept
Vector equation of a line passing through $\vec{a}$ and $\vec{b}$ is $\vec{r} = \vec{a} + \lambda(\vec{b}-\vec{a})$.

Step 2: Meaning
$\vec{a} = (0,0,1) = \hat{k}$ and $\vec{b} = (1,1,0) = \hat{i}+\hat{j}$.

Step 3: Analysis
Direction vector $\vec{d} = \vec{b} - \vec{a} = (\hat{i}+\hat{j}) - \hat{k} = \hat{i}+\hat{j}-\hat{k}$.

Step 4: Conclusion
The equation is $\vec{r} = \hat{k} + \lambda(\hat{i}+\hat{j}-\hat{k})$. Final Answer: (A)