If `vec(A) = 3 hati +2hatj` and `vec(B) = hati - 2hatj +3hatk`, find the magnitudes of `vec(A) +vec(B)` and `vec(A) - vec(B)`.

Answer

To solve the problem, we need to find the magnitudes of the vectors \(\vec{A} + \vec{B}\) and \(\vec{A} - \vec{B}\). We start with the given vectors:

\[ \vec{A} = 3 \hat{i} + 2 \hat{j} \]
\[ \vec{B} = \hat{i} - 2 \hat{j} + 3 \hat{k} \]

### Step 1: Calculate \(\vec{A} + \vec{B}\)

To find \(\vec{A} + \vec{B}\), we add the corresponding components of the vectors:

\[ \vec{A} + \vec{B} = (3 \hat{i} + 2 \hat{j}) + (\hat{i} - 2 \hat{j} + 3 \hat{k}) \]

Combining the components:

\[ \vec{A} + \vec{B} = (3 + 1) \hat{i} + (2 - 2) \hat{j} + (0 + 3) \hat{k} = 4 \hat{i} + 0 \hat{j} + 3 \hat{k} \]

Thus,

\[ \vec{A} + \vec{B} = 4 \hat{i} + 3 \hat{k} \]

### Step 2: Calculate the magnitude of \(\vec{A} + \vec{B}\)

The magnitude of a vector \(\vec{C} = x \hat{i} + y \hat{j} + z \hat{k}\) is given by:

\[ |\vec{C}| = \sqrt{x^2 + y^2 + z^2} \]

For \(\vec{A} + \vec{B} = 4 \hat{i} + 0 \hat{j} + 3 \hat{k}\):

\[ |\vec{A} + \vec{B}| = \sqrt{4^2 + 0^2 + 3^2} = \sqrt{16 + 0 + 9} = \sqrt{25} = 5 \]

### Step 3: Calculate \(\vec{A} - \vec{B}\)

Now, we find \(\vec{A} - \vec{B}\):

\[ \vec{A} - \vec{B} = (3 \hat{i} + 2 \hat{j}) - (\hat{i} - 2 \hat{j} + 3 \hat{k}) \]

Combining the components:

\[ \vec{A} - \vec{B} = (3 - 1) \hat{i} + (2 + 2) \hat{j} + (0 - 3) \hat{k} = 2 \hat{i} + 4 \hat{j} - 3 \hat{k} \]

Thus,

\[ \vec{A} - \vec{B} = 2 \hat{i} + 4 \hat{j} - 3 \hat{k} \]

### Step 4: Calculate the magnitude of \(\vec{A} - \vec{B}\)

Using the magnitude formula again:

\[ |\vec{A} - \vec{B}| = \sqrt{2^2 + 4^2 + (-3)^2} = \sqrt{4 + 16 + 9} = \sqrt{29} \]

### Final Results

The magnitudes are:

1. \(|\vec{A} + \vec{B}| = 5\)
2. \(|\vec{A} - \vec{B}| = \sqrt{29}\)