`Let `veca=4veci+3vecj and vecb=3veci+4vecj`. a.Find the magnitudes of a. veca, b. `vecb, c. veca+vecb and d. veca-vecb.`

Explanation

To solve the problem step by step, we will find the magnitudes of the vectors \(\vec{a}\), \(\vec{b}\), \(\vec{a} + \vec{b}\), and \(\vec{a} - \vec{b}\).

### Step 1: Find the magnitude of \(\vec{a}\)

Given:
\[ \vec{a} = 4\hat{i} + 3\hat{j} \]

The magnitude of a vector \(\vec{v} = x\hat{i} + y\hat{j}\) is given by:
\[ |\vec{v}| = \sqrt{x^2 + y^2} \]

For \(\vec{a}\):
\[ |\vec{a}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 2: Find the magnitude of \(\vec{b}\)

Given:
\[ \vec{b} = 3\hat{i} + 4\hat{j} \]

Using the same formula for magnitude:
\[ |\vec{b}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 3: Find the magnitude of \(\vec{a} + \vec{b}\)

First, we need to add the vectors:
\[ \vec{a} + \vec{b} = (4\hat{i} + 3\hat{j}) + (3\hat{i} + 4\hat{j}) = (4 + 3)\hat{i} + (3 + 4)\hat{j} = 7\hat{i} + 7\hat{j} \]

Now, find the magnitude:
\[ |\vec{a} + \vec{b}| = \sqrt{7^2 + 7^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2} \]

### Step 4: Find the magnitude of \(\vec{a} - \vec{b}\)

Now, we subtract the vectors:
\[ \vec{a} - \vec{b} = (4\hat{i} + 3\hat{j}) - (3\hat{i} + 4\hat{j}) = (4 - 3)\hat{i} + (3 - 4)\hat{j} = 1\hat{i} - 1\hat{j} \]

Now, find the magnitude:
\[ |\vec{a} - \vec{b}| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Final Results
- Magnitude of \(\vec{a}\): \(5\)
- Magnitude of \(\vec{b}\): \(5\)
- Magnitude of \(\vec{a} + \vec{b}\): \(7\sqrt{2}\)
- Magnitude of \(\vec{a} - \vec{b}\): \(\sqrt{2}\)