Question

Which one of the following is a vector parallel to the straight line $\vec{r}=(\hat{i}-11\hat{j}+101\hat{k})+\lambda(3\hat{i}-5\hat{j}+2\hat{k}),\lambda\in\mathbb{R}$? ________.

• A. $-3\hat{i}+5\hat{j}-2\hat{k}$
• B. $3\hat{i}+5\hat{j}+2\hat{k}$
• C. $\hat{i}-11\hat{j}+101\hat{k}$
• D. $-\hat{i}+11\hat{j}+101\hat{k}$
• E. $-4\hat{i}-16\hat{j}+103\hat{k}$

Hint:

A vector $\vec{v}$ is parallel to $\vec{d}$ if $\vec{v} = k\vec{d}$.

Answer 1

The Correct Option is A

Step 1: Concept
In the line equation $\vec{r} = \vec{a} + \lambda\vec{d}$, $\vec{d}$ is the direction vector.

Step 2: Meaning
The direction vector of the line is $3\hat{i}-5\hat{j}+2\hat{k}$.

Step 3: Analysis
Any vector parallel to the line must be a multiple of this direction vector.

Step 4: Conclusion
$-1 \times (3\hat{i}-5\hat{j}+2\hat{k}) = -3\hat{i}+5\hat{j}-2\hat{k}$. Final Answer: (A)