Question

If the vector \(\bar{r}\) is perpendicular to \(\hat{i}-\hat{k}\) and \(\bar{r}\times (\hat{i}+x\hat{j}-\hat{k})=\hat{i}+2\hat{j}+3\hat{k}\), then the value of \(\bar{r}\cdot (\hat{i}+x\hat{j}-\hat{k})=\)

• A. \(-2\)
• B. \(-1\)
• C. \(1\)
• D. \(2\)

Hint:

Use that r x v is perpendicular to v to find x, then use r perpendicular to i - k to solve for r.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Let \(\bar{v}=\hat{i}+x\hat{j}-\hat{k}\). The vector \(\bar{r}\times\bar{v}\) is perpendicular to \(\bar{v}\).

Step 2: Find x
\((\hat{i}+2\hat{j}+3\hat{k})\cdot(\hat{i}+x\hat{j}-\hat{k})=0\Rightarrow 1+2x-3=0\Rightarrow x=1\).

Step 3: Find r
\(\bar{r}\perp(\hat{i}-\hat{k})\) gives \(\bar{r}=(p,q,p)\).
\(\bar{r}\times(1,1,-1)=(-q-p,\ 2p,\ p-q)=(1,2,3)\), so \(p=1\), \(q=-2\).

Step 4: Calculation
\(\bar{r}=\hat{i}-2\hat{j}+\hat{k}\).
\(\bar{r}\cdot(\hat{i}+\hat{j}-\hat{k})=1-2-1=-2\).

Final Answer:
The dot product equals -2. \[ \boxed{\text{(A) }-2} \]