If `|vec(a)|=7, |vec(b)|=11 and |vec(a)+vec(b)|=10 sqrt(3)`, then `|vec(a)-vec(b)|` is equal to

Options

A. 40
B. 10
C. `4sqrt(10)`
D. `2sqrt(10)`

Explanation

To solve the problem, we need to find the magnitude of \(|\vec{a} - \vec{b}|\) given the following information:

1. \(|\vec{a}| = 7\)
2. \(|\vec{b}| = 11\)
3. \(|\vec{a} + \vec{b}| = 10\sqrt{3}\)

### Step 1: Use the formula for the magnitude of the sum of two vectors
The formula for the magnitude of the sum of two vectors is given by:

\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos\theta \]

Where \(\theta\) is the angle between the vectors \(\vec{a}\) and \(\vec{b}\).

### Step 2: Substitute the known values into the equation
We know:

- \(|\vec{a}|^2 = 7^2 = 49\)
- \(|\vec{b}|^2 = 11^2 = 121\)
- \(|\vec{a} + \vec{b}|^2 = (10\sqrt{3})^2 = 300\)

Now substituting these values into the equation:

\[ 300 = 49 + 121 + 2 \cdot 7 \cdot 11 \cdot \cos\theta \]

### Step 3: Simplify the equation
Combine the constants:

\[ 300 = 170 + 154\cos\theta \]

Now isolate \(\cos\theta\):

\[ 300 - 170 = 154\cos\theta \]
\[ 130 = 154\cos\theta \]
\[ \cos\theta = \frac{130}{154} = \frac{65}{77} \]

### Step 4: Use the formula for the magnitude of the difference of two vectors
The formula for the magnitude of the difference of two vectors is given by:

\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2|\vec{a}||\vec{b}|\cos\theta \]

### Step 5: Substitute the known values into the equation
Substituting the values we have:

\[ |\vec{a} - \vec{b}|^2 = 49 + 121 - 2 \cdot 7 \cdot 11 \cdot \frac{65}{77} \]

### Step 6: Calculate the components
First, calculate \(49 + 121\):

\[ 49 + 121 = 170 \]

Now calculate \(2 \cdot 7 \cdot 11 \cdot \frac{65}{77}\):

\[ 2 \cdot 7 \cdot 11 = 154 \]
\[ 154 \cdot \frac{65}{77} = 130 \]

### Step 7: Substitute back into the equation
Now we can substitute back:

\[ |\vec{a} - \vec{b}|^2 = 170 - 130 = 40 \]

### Step 8: Take the square root to find the magnitude
Finally, take the square root to find \(|\vec{a} - \vec{b}|\):

\[ |\vec{a} - \vec{b}| = \sqrt{40} = 2\sqrt{10} \]

### Final Answer
Thus, the magnitude \(|\vec{a} - \vec{b}|\) is equal to \(2\sqrt{10}\).

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