Question

Let \(\overset{⃗}{a} = 2\hat{i}+\hat{k},\overset{⃗}{b} = \hat{i}+\hat{j}+\hat{k}\), and \(\overset{⃗}{c} = 4\hat{i}-3\hat{j}+7\hat{k}\). If \(\overset{⃗}{r}\) is a vector such that \(\overset{⃗}{r}\times \overset{⃗}{b} = \overset{⃗}{c}\times \overset{⃗}{b}\) and \(\overset{⃗}{r}\cdot \overset{⃗}{a} = 0\), Then \(\overset{⃗}{r}\cdot \overset{⃗}{c} =\)

• A. \(-14\)
• B. \(34\)
• C. \(-7\)
• D. \(20\)

Hint:

From r cross b = c cross b we get r minus c parallel to b.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
\(\vec r\times\vec b=\vec c\times\vec b\) means \((\vec r-\vec c)\times\vec b=\vec 0\). So \(\vec r-\vec c\) is parallel to \(\vec b\).

Step 2: Write r
\[ \vec r=\vec c+\lambda\vec b \]

Step 3: Use the dot condition
\[ \vec r\cdot\vec a=0\Rightarrow\vec c\cdot\vec a+\lambda\,\vec b\cdot\vec a=0 \]
\(\vec c\cdot\vec a=8+7=15\) and \(\vec b\cdot\vec a=2+1=3\). So \(15+3\lambda=0\), \(\lambda=-5\).

Step 4: Find r
\[ \vec r=(4,-3,7)-5(1,1,1)=(-1,-8,2) \]

Step 5: Required dot product
\[ \vec r\cdot\vec c=(-1)(4)+(-8)(-3)+(2)(7)=-4+24+14=34 \]
This is option (B).

Final Answer:
The vector r equals c - 5b, and its dot product with c is 34, option (B). \[ \boxed{34} \]