Find `| vec a- vec b|,`if two vector ` vec a`and ` vec b`are such that `| vec a|=2,| vec b|=3`and ` vec adot vec b=4`.

Options

A. `sqrt5`
B. 5
C. 2
D. `sqrt2`

Explanation

To find the magnitude of the vector \( \vec{a} - \vec{b} \), we can use the formula for the magnitude of the difference of two vectors. The formula is given by:

\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2(\vec{a} \cdot \vec{b}) \]

### Step 1: Write down the known values
We are given:
- \( |\vec{a}| = 2 \)
- \( |\vec{b}| = 3 \)
- \( \vec{a} \cdot \vec{b} = 4 \)

### Step 2: Substitute the known values into the formula
Substituting the values into the formula gives:

\[ |\vec{a} - \vec{b}|^2 = (2)^2 + (3)^2 - 2(4) \]

### Step 3: Calculate the squares and products
Now we calculate each term:

- \( (2)^2 = 4 \)
- \( (3)^2 = 9 \)
- \( 2(4) = 8 \)

So we can rewrite the equation as:

\[ |\vec{a} - \vec{b}|^2 = 4 + 9 - 8 \]

### Step 4: Simplify the equation
Now, simplify the right-hand side:

\[ |\vec{a} - \vec{b}|^2 = 4 + 9 - 8 = 5 \]

### Step 5: Take the square root
To find \( |\vec{a} - \vec{b}| \), we take the square root of both sides:

\[ |\vec{a} - \vec{b}| = \sqrt{5} \]

### Final Answer
Thus, the magnitude of the vector \( \vec{a} - \vec{b} \) is:

\[ |\vec{a} - \vec{b}| = \sqrt{5} \]