Vectors `ahati+bhat j +hatk` and `2hati -3hatj+4hatk` are perpendicular to each other when 3a+2b=7,the radio of a and b is `x/2`. The value of is____.

Answer

To solve the problem, we need to determine the values of \( a \) and \( b \) such that the vectors \( \hat{i} a + \hat{j} b + \hat{k} \) and \( 2\hat{i} - 3\hat{j} + 4\hat{k} \) are perpendicular to each other, given that \( 3a + 2b = 7 \) and the ratio \( \frac{a}{b} = \frac{x}{2} \).

### Step 1: Set up the dot product condition for perpendicular vectors
Two vectors are perpendicular if their dot product is zero. Therefore, we need to calculate the dot product of the two vectors:
\[ (\hat{i} a + \hat{j} b + \hat{k}) \cdot (2\hat{i} - 3\hat{j} + 4\hat{k}) = 0 \]
Calculating the dot product:
\[ 2a - 3b + 4 = 0 \]
This gives us our second equation.

### Step 2: Write down the equations
We now have two equations:
1. \( 3a + 2b = 7 \) (Equation 1)
2. \( 2a - 3b + 4 = 0 \) (Equation 2)

### Step 3: Rearranging Equation 2
Rearranging Equation 2:
\[ 2a - 3b = -4 \]
This can be rewritten as:
\[ 2a = 3b - 4 \]
(Equation 3)

### Step 4: Substitute Equation 3 into Equation 1
Now, we can substitute \( 2a \) from Equation 3 into Equation 1. First, we express \( a \) in terms of \( b \):
From Equation 3:
\[ a = \frac{3b - 4}{2} \]
Substituting this into Equation 1:
\[ 3\left(\frac{3b - 4}{2}\right) + 2b = 7 \]
Multiplying through by 2 to eliminate the fraction:
\[ 3(3b - 4) + 4b = 14 \]
Expanding:
\[ 9b - 12 + 4b = 14 \]
Combining like terms:
\[ 13b - 12 = 14 \]
Adding 12 to both sides:
\[ 13b = 26 \]
Dividing by 13:
\[ b = 2 \]

### Step 5: Substitute back to find \( a \)
Now substitute \( b = 2 \) back into Equation 1 to find \( a \):
\[ 3a + 2(2) = 7 \]
This simplifies to:
\[ 3a + 4 = 7 \]
Subtracting 4 from both sides:
\[ 3a = 3 \]
Dividing by 3:
\[ a = 1 \]

### Step 6: Calculate the ratio \( \frac{a}{b} \)
Now we have \( a = 1 \) and \( b = 2 \). The ratio \( \frac{a}{b} \) is:
\[ \frac{a}{b} = \frac{1}{2} \]

### Step 7: Relate to \( \frac{x}{2} \)
Given that \( \frac{a}{b} = \frac{x}{2} \), we can equate:
\[ \frac{1}{2} = \frac{x}{2} \]
Cross-multiplying gives:
\[ 1 = x \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{1} \]