If `|veca + vecb| =60 |veca-vecb|=40 and |vecb| = 46` find `|veca|`

Answer

To solve the problem step by step, we will use the given information about the magnitudes of the vectors and apply the properties of vector addition and subtraction.

### Step 1: Write down the equations based on the given information.
We are given:
1. \( |\vec{a} + \vec{b}| = 60 \)
2. \( |\vec{a} - \vec{b}| = 40 \)
3. \( |\vec{b}| = 46 \)

### Step 2: Square both sides of the first equation.
From the first equation, we can square both sides:
\[ |\vec{a} + \vec{b}|^2 = 60^2 \]
This gives us:
\[ |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} = 3600 \tag{1} \]

### Step 3: Square both sides of the second equation.
Now, squaring the second equation:
\[ |\vec{a} - \vec{b}|^2 = 40^2 \]
This results in:
\[ |\vec{a}|^2 + |\vec{b}|^2 - 2 \vec{a} \cdot \vec{b} = 1600 \tag{2} \]

### Step 4: Add the two equations (1) and (2).
Adding equations (1) and (2):
\[ \left( |\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b} \right) + \left( |\vec{a}|^2 + |\vec{b}|^2 - 2 \vec{a} \cdot \vec{b} \right) = 3600 + 1600 \]
This simplifies to:
\[ 2|\vec{a}|^2 + 2|\vec{b}|^2 = 5200 \]
Dividing by 2:
\[ |\vec{a}|^2 + |\vec{b}|^2 = 2600 \tag{3} \]

### Step 5: Substitute the value of \( |\vec{b}| \).
We know \( |\vec{b}| = 46 \), so we can substitute this into equation (3):
\[ |\vec{a}|^2 + 46^2 = 2600 \]
Calculating \( 46^2 \):
\[ 46^2 = 2116 \]
Thus, we have:
\[ |\vec{a}|^2 + 2116 = 2600 \]

### Step 6: Solve for \( |\vec{a}|^2 \).
Now, we isolate \( |\vec{a}|^2 \):
\[ |\vec{a}|^2 = 2600 - 2116 \]
Calculating the right side:
\[ |\vec{a}|^2 = 484 \]

### Step 7: Take the square root to find \( |\vec{a}| \).
Finally, we take the square root:
\[ |\vec{a}| = \sqrt{484} = 22 \]

### Conclusion:
The magnitude of vector \( \vec{a} \) is \( 22 \).

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