Vector Algebra
Lines in Space
Grade None
Question:
<p>A line passes through point \(A\) with position vector \(\vec{v}=\hat{j}-\hat{k}\) and is parallel to the vector \(\hat{i}+\hat{j}\). For any point \(P\) on this line, which of the following is/are true?</p>
<li>\(\overrightarrow{AP}\parallel(\hat{i}+\hat{j})\)</li>
<li>Position vector of P: \((\hat{j}-\hat{k})+t(\hat{i}+\hat{j}),\;t\in\mathbb{R}\)</li>
<li>\(\overrightarrow{AP}=t(\hat{i}+\hat{j})\) for some \(t\)</li>
<li>\(\vec{r}=(\hat{j}-\hat{k})+\lambda(\hat{i}+\hat{j})\) is the line equation</li>
Step-by-Step Solution
Key Concept: The parametric line equation is r = a + td where a is position of A and d is direction. All points P satisfy AP = td.
Line: $\vec{r}=(\hat{j}-\hat{k})+t(\hat{i}+\hat{j})$, $t\in\mathbb{R}$.
For any P on line: $\overrightarrow{AP}=\vec{r}-\vec{v}=t(\hat{i}+\hat{j})\Rightarrow \overrightarrow{AP}\parallel(\hat{i}+\hat{j})$. ✓ (A, C, D)
Answer: ACD
Correct Answer: ACD