If a and b are two non-collinear vectors such that `|a| = 3,|b| = 4 and a - b= hati +2hatj + 3hatk` , then the value of `{abs(a-b)/(abs(a)abs(b))}^2`

Options

A. `1/24`
B. `5/72`
C. `7/72`
D. `7/48`

Answer

To solve the problem, we need to find the value of \(\left(\frac{|a - b|}{|a||b|}\right)^2\).

### Step 1: Find \(|a - b|\)
We know that:
\[ a - b = \hat{i} + 2\hat{j} + 3\hat{k} \]
To find \(|a - b|\), we calculate the magnitude:
\[ |a - b| = \sqrt{(1)^2 + (2)^2 + (3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Step 2: Calculate \(|a|\) and \(|b|\)
We are given:
\[ |a| = 3 \quad \text{and} \quad |b| = 4 \]

### Step 3: Calculate \(|a||b|\)
Now, we calculate the product of the magnitudes:
\[ |a||b| = |a| \cdot |b| = 3 \cdot 4 = 12 \]

### Step 4: Substitute into the expression
Now, we substitute \(|a - b|\) and \(|a||b|\) into the expression:
\[ \frac{|a - b|}{|a||b|} = \frac{\sqrt{14}}{12} \]

### Step 5: Square the result
Now we need to square the result:
\[ \left(\frac{|a - b|}{|a||b|}\right)^2 = \left(\frac{\sqrt{14}}{12}\right)^2 = \frac{14}{144} = \frac{7}{72} \]

### Final Result
Thus, the value of \(\left(\frac{|a - b|}{|a||b|}\right)^2\) is:
\[ \frac{7}{72} \]