IF `barA=hati-3hatj+4hatk,barB=6hati+4hatj-8hatk,barC=5hati+2hatj+5hatk` and a vector `barR` satisfies `barR times barB=barC times barB,barR. barA=0` then `|barB|/(|barR-barC|)` is equal to

Options

A. 1
B. `1//2`
C. 2
D. 3

Answer

To solve the problem, we need to find the value of \(\frac{|\bar{B}|}{|\bar{R} - \bar{C}|}\) given the vectors \(\bar{A}\), \(\bar{B}\), and \(\bar{C}\) and the conditions on \(\bar{R}\).

### Step 1: Identify the vectors
We have:
\[ \bar{A} = \hat{i} - 3\hat{j} + 4\hat{k} \]
\[ \bar{B} = 6\hat{i} + 4\hat{j} - 8\hat{k} \]
\[ \bar{C} = 5\hat{i} + 2\hat{j} + 5\hat{k} \]

### Step 2: Calculate \(|\bar{B}|\)
The magnitude of vector \(\bar{B}\) is calculated as follows:
\[ |\bar{B}| = \sqrt{(6)^2 + (4)^2 + (-8)^2} = \sqrt{36 + 16 + 64} = \sqrt{116} = 2\sqrt{29} \]

### Step 3: Use the condition \(\bar{R} \times \bar{B} = \bar{C} \times \bar{B}\)
This condition implies that \(\bar{R}\) lies in the plane defined by \(\bar{B}\) and \(\bar{C}\). We can express \(\bar{R}\) in terms of \(\bar{C}\) and a scalar multiple of \(\bar{B}\):
\[ \bar{R} = \bar{C} + k\bar{B} \]
for some scalar \(k\).

### Step 4: Use the condition \(\bar{R} \cdot \bar{A} = 0\)
Substituting \(\bar{R}\) into this condition:
\[ (\bar{C} + k\bar{B}) \cdot \bar{A} = 0 \]
Expanding this gives:
\[ \bar{C} \cdot \bar{A} + k(\bar{B} \cdot \bar{A}) = 0 \]
From this, we can solve for \(k\):
\[ k = -\frac{\bar{C} \cdot \bar{A}}{\bar{B} \cdot \bar{A}} \]

### Step 5: Calculate \(\bar{C} \cdot \bar{A}\) and \(\bar{B} \cdot \bar{A}\)
Calculating \(\bar{C} \cdot \bar{A}\):
\[ \bar{C} \cdot \bar{A} = (5)(1) + (2)(-3) + (5)(4) = 5 - 6 + 20 = 19 \]

Calculating \(\bar{B} \cdot \bar{A}\):
\[ \bar{B} \cdot \bar{A} = (6)(1) + (4)(-3) + (-8)(4) = 6 - 12 - 32 = -38 \]

### Step 6: Substitute back to find \(k\)
Substituting these values into the equation for \(k\):
\[ k = -\frac{19}{-38} = \frac{1}{2} \]

### Step 7: Find \(\bar{R}\)
Now substituting \(k\) back into the expression for \(\bar{R}\):
\[ \bar{R} = \bar{C} + \frac{1}{2}\bar{B} = (5\hat{i} + 2\hat{j} + 5\hat{k}) + \frac{1}{2}(6\hat{i} + 4\hat{j} - 8\hat{k}) \]
Calculating this gives:
\[ \bar{R} = 5\hat{i} + 2\hat{j} + 5\hat{k} + 3\hat{i} + 2\hat{j} - 4\hat{k} = (5 + 3)\hat{i} + (2 + 2)\hat{j} + (5 - 4)\hat{k} = 8\hat{i} + 4\hat{j} + 1\hat{k} \]

### Step 8: Calculate \(|\bar{R} - \bar{C}|\)
Now we find \(\bar{R} - \bar{C}\):
\[ \bar{R} - \bar{C} = (8\hat{i} + 4\hat{j} + 1\hat{k}) - (5\hat{i} + 2\hat{j} + 5\hat{k}) = (8 - 5)\hat{i} + (4 - 2)\hat{j} + (1 - 5)\hat{k} = 3\hat{i} + 2\hat{j} - 4\hat{k} \]
Calculating the magnitude:
\[ |\bar{R} - \bar{C}| = \sqrt{(3)^2 + (2)^2 + (-4)^2} = \sqrt{9 + 4 + 16} = \sqrt{29} \]

### Step 9: Calculate \(\frac{|\bar{B}|}{|\bar{R} - \bar{C}|}\)
Now substituting the magnitudes:
\[ \frac{|\bar{B}|}{|\bar{R} - \bar{C}|} = \frac{2\sqrt{29}}{\sqrt{29}} = 2 \]

### Final Answer
Thus, the value of \(\frac{|\bar{B}|}{|\bar{R} - \bar{C}|}\) is \(2\).