If `|veca+vecb|=60,|veca-vecb|=40 and |veca|=22` then find `|vecb|.`

Options

A. `vec|b|=46`
B. `vec|b|=48`
C. `vec|b|=44`
D. `vec|b|=50`

Answer

To solve the problem step by step, we will use the information given about the vectors \(\vec{a}\) and \(\vec{b}\).

### Step 1: Write down the given information
We have the following information:
- \(|\vec{a} + \vec{b}| = 60\)
- \(|\vec{a} - \vec{b}| = 40\)
- \(|\vec{a}| = 22\)

### Step 2: Square the magnitudes
We will square the magnitudes of the vectors:
1. \(|\vec{a} + \vec{b}|^2 = 60^2 = 3600\)
2. \(|\vec{a} - \vec{b}|^2 = 40^2 = 1600\)

### Step 3: Use the formula for the square of the magnitude of vector sums
Using the formula:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} \]
we can write:
\[ |\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} = 3600 \quad \text{(Equation 1)} \]

### Step 4: Use the formula for the square of the magnitude of vector differences
Similarly, for the difference:
\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2 \vec{a} \cdot \vec{b} \]
we can write:
\[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2 \vec{a} \cdot \vec{b} = 1600 \quad \text{(Equation 2)} \]

### Step 5: Set up the equations
From Equation 1:
\[ |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} = 3600 \]
From Equation 2:
\[ |\vec{a}|^2 + |\vec{b}|^2 - 2 \vec{a} \cdot \vec{b} = 1600 \]

### Step 6: Add the two equations
Adding Equation 1 and Equation 2:
\[ 2(|\vec{a}|^2 + |\vec{b}|^2) = 3600 + 1600 = 5200 \]
Thus,
\[ |\vec{a}|^2 + |\vec{b}|^2 = \frac{5200}{2} = 2600 \quad \text{(Equation 3)} \]

### Step 7: Substitute the known value of \(|\vec{a}|\)
We know \(|\vec{a}| = 22\), so:
\[ |\vec{a}|^2 = 22^2 = 484 \]
Substituting this into Equation 3:
\[ 484 + |\vec{b}|^2 = 2600 \]

### Step 8: Solve for \(|\vec{b}|^2\)
Rearranging gives:
\[ |\vec{b}|^2 = 2600 - 484 = 2116 \]

### Step 9: Find \(|\vec{b}|\)
Taking the square root:
\[ |\vec{b}| = \sqrt{2116} = 46 \]

### Final Answer
Thus, the magnitude of vector \(\vec{b}\) is:
\[ |\vec{b}| = 46 \]