A, B and C are three vectors given by `2hat(i)+hat(k), hat(i)+hat(j)+hat(k) and 4hat(i)-3hat(j)+7hat(k)`. Then, find R, which satisfies the relation `RtimesB=CtimesB and R*A=0`.

Answer

To solve the problem, we need to find the vector \( R \) that satisfies the conditions \( R \times B = C \times B \) and \( R \cdot A = 0 \). Let's break down the solution step by step.

### Given Vectors:
1. \( A = 2\hat{i} + \hat{k} \)
2. \( B = \hat{i} + \hat{j} + \hat{k} \)
3. \( C = 4\hat{i} - 3\hat{j} + 7\hat{k} \)

### Step 1: Set Up the Vector \( R \)
Assume \( R \) can be expressed as:
\[ R = x\hat{i} + y\hat{j} + z\hat{k} \]

### Step 2: Use the Condition \( R \cdot A = 0 \)
Using the dot product condition:
\[ R \cdot A = (x\hat{i} + y\hat{j} + z\hat{k}) \cdot (2\hat{i} + \hat{k}) = 0 \]
Calculating the dot product:
\[ 2x + 0y + z = 0 \implies 2x + z = 0 \tag{1} \]

### Step 3: Use the Condition \( R \times B = C \times B \)
From the second condition:
\[ R \times B = C \times B \]
This implies:
\[ R \times B - C \times B = 0 \implies (R - C) \times B = 0 \]
This means \( R - C \) is parallel to \( B \). Therefore, we can express \( R \) as:
\[ R = C + kB \quad \text{for some scalar } k \]

### Step 4: Substitute \( C \) and \( B \)
Substituting the values of \( C \) and \( B \):
\[ R = (4\hat{i} - 3\hat{j} + 7\hat{k}) + k(\hat{i} + \hat{j} + \hat{k}) \]
Expanding this gives:
\[ R = (4 + k)\hat{i} + (-3 + k)\hat{j} + (7 + k)\hat{k} \]

### Step 5: Equate to the General Form of \( R \)
Now, we can compare coefficients:
\[ x = 4 + k, \quad y = -3 + k, \quad z = 7 + k \]

### Step 6: Substitute into Equation (1)
Substituting into equation (1):
\[ 2(4 + k) + (7 + k) = 0 \]
Expanding this:
\[ 8 + 2k + 7 + k = 0 \implies 15 + 3k = 0 \implies k = -5 \]

### Step 7: Find \( x, y, z \)
Substituting \( k = -5 \) back into the equations for \( x, y, z \):
\[ x = 4 - 5 = -1 \]
\[ y = -3 - 5 = -8 \]
\[ z = 7 - 5 = 2 \]

### Step 8: Write the Final Vector \( R \)
Thus, the vector \( R \) is:
\[ R = -1\hat{i} - 8\hat{j} + 2\hat{k} \]

### Final Answer:
\[ R = -\hat{i} - 8\hat{j} + 2\hat{k} \]