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 equation involving two perpendicular vectors \(\vec{a}\) and \(\vec{b}\) of equal magnitude, and we need to find the magnitude of \(\vec{a}\).

### Step 1: Understand the given information
We know that \(|\vec{a}| = |\vec{b}|\) and that \(\vec{a}\) and \(\vec{b}\) are perpendicular. Let's denote the magnitude of both vectors as \(k\), so we have:
\[ |\vec{a}| = |\vec{b}| = k \]

### Step 2: Express the vectors
Since \(\vec{a}\) and \(\vec{b}\) are perpendicular, we can represent them in a 2D plane as:
\[ \vec{a} = k \hat{i} \quad \text{and} \quad \vec{b} = k \hat{j} \]
where \(\hat{i}\) and \(\hat{j}\) are unit vectors along the x-axis and y-axis, respectively.

### Step 3: Substitute into the equation
We need to evaluate the expression \(|3\vec{a} + 4\vec{b}| + |4\vec{a} - 3\vec{b}|\):
1. Calculate \(3\vec{a} + 4\vec{b}\):
\[ 3\vec{a} + 4\vec{b} = 3(k \hat{i}) + 4(k \hat{j}) = 3k \hat{i} + 4k \hat{j} \]
The magnitude is:
\[ |3\vec{a} + 4\vec{b}| = \sqrt{(3k)^2 + (4k)^2} = \sqrt{9k^2 + 16k^2} = \sqrt{25k^2} = 5k \]

2. Calculate \(4\vec{a} - 3\vec{b}\):
\[ 4\vec{a} - 3\vec{b} = 4(k \hat{i}) - 3(k \hat{j}) = 4k \hat{i} - 3k \hat{j} \]
The magnitude is:
\[ |4\vec{a} - 3\vec{b}| = \sqrt{(4k)^2 + (-3k)^2} = \sqrt{16k^2 + 9k^2} = \sqrt{25k^2} = 5k \]

### Step 4: Combine the magnitudes
Now we combine the magnitudes:
\[ |3\vec{a} + 4\vec{b}| + |4\vec{a} - 3\vec{b}| = 5k + 5k = 10k \]

### Step 5: Set up the equation
According to the problem statement, we have:
\[ 10k = 20 \]

### Step 6: Solve for \(k\)
Now, we solve for \(k\):
\[ k = \frac{20}{10} = 2 \]

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