If `veca,vecb` are any two perpendicular vectors of equal magnitude and `|3veca+4vecb|+|4veca-3vecb|=20` then `|veca|` equals :

Explanation

To solve the problem, we start with the given information about the vectors \(\vec{a}\) and \(\vec{b}\).

### Step 1: Define the Magnitudes
Let the magnitude of both vectors \(\vec{a}\) and \(\vec{b}\) be \(k\). Therefore, we have:
\[ |\vec{a}| = |\vec{b}| = k \]

### Step 2: Use the Property of Perpendicular Vectors
Since \(\vec{a}\) and \(\vec{b}\) are perpendicular, we know:
\[ \vec{a} \cdot \vec{b} = 0 \]

### Step 3: Calculate the Magnitude of \(3\vec{a} + 4\vec{b}\)
We need to find the magnitude of the vector \(3\vec{a} + 4\vec{b}\):
\[ |3\vec{a} + 4\vec{b}|^2 = |3\vec{a}|^2 + |4\vec{b}|^2 + 2(\vec{a} \cdot \vec{b}) \cdot 3 \cdot 4 \]
Since \(\vec{a} \cdot \vec{b} = 0\), this simplifies to:
\[ |3\vec{a} + 4\vec{b}|^2 = (3k)^2 + (4k)^2 = 9k^2 + 16k^2 = 25k^2 \]
Thus,
\[ |3\vec{a} + 4\vec{b}| = 5k \]

### Step 4: Calculate the Magnitude of \(4\vec{a} - 3\vec{b}\)
Next, we calculate the magnitude of the vector \(4\vec{a} - 3\vec{b}\):
\[ |4\vec{a} - 3\vec{b}|^2 = |4\vec{a}|^2 + |-3\vec{b}|^2 + 2(\vec{a} \cdot \vec{b}) \cdot 4 \cdot (-3) \]
Again, since \(\vec{a} \cdot \vec{b} = 0\), this simplifies to:
\[ |4\vec{a} - 3\vec{b}|^2 = (4k)^2 + (-3k)^2 = 16k^2 + 9k^2 = 25k^2 \]
Thus,
\[ |4\vec{a} - 3\vec{b}| = 5k \]

### Step 5: Combine the Results
Now we combine the magnitudes we calculated:
\[ |3\vec{a} + 4\vec{b}| + |4\vec{a} - 3\vec{b}| = 5k + 5k = 10k \]
According to the problem statement, this equals 20:
\[ 10k = 20 \]

### Step 6: Solve for \(k\)
Dividing both sides by 10 gives:
\[ k = 2 \]

### Conclusion
Thus, the magnitude of \(\vec{a}\) is:
\[ |\vec{a}| = k = 2 \]