If `|vec(a)|=3, |vec(b)|=4 and |vec(a)-vec(b)|=5`, then what is the value of `|vec(a)+vec(b)|=?`

Options

A. 8
B. 6
C. `5sqrt(2)`
D. 5

Explanation

To solve the problem, we need to find the value of \(|\vec{a} + \vec{b}|\) given the following information:

1. \(|\vec{a}| = 3\)
2. \(|\vec{b}| = 4\)
3. \(|\vec{a} - \vec{b}| = 5\)

We can use the following formula that relates the magnitudes of vectors:

\[ |\vec{a} + \vec{b}|^2 + |\vec{a} - \vec{b}|^2 = 2(|\vec{a}|^2 + |\vec{b}|^2) \]

### Step 1: Calculate \(|\vec{a}|^2\) and \(|\vec{b}|^2\)

\[ |\vec{a}|^2 = 3^2 = 9 \]
\[ |\vec{b}|^2 = 4^2 = 16 \]

### Step 2: Calculate \(|\vec{a}|^2 + |\vec{b}|^2\)

\[ |\vec{a}|^2 + |\vec{b}|^2 = 9 + 16 = 25 \]

### Step 3: Calculate \(|\vec{a} - \vec{b}|^2\)

\[ |\vec{a} - \vec{b}|^2 = 5^2 = 25 \]

### Step 4: Substitute into the formula

Now, substitute the values into the formula:

\[ |\vec{a} + \vec{b}|^2 + 25 = 2 \times 25 \]

### Step 5: Simplify the equation

\[ |\vec{a} + \vec{b}|^2 + 25 = 50 \]
\[ |\vec{a} + \vec{b}|^2 = 50 - 25 \]
\[ |\vec{a} + \vec{b}|^2 = 25 \]

### Step 6: Take the square root

Now, take the square root to find \(|\vec{a} + \vec{b}|\):

\[ |\vec{a} + \vec{b}| = \sqrt{25} = 5 \]

### Final Answer

Thus, the value of \(|\vec{a} + \vec{b}|\) is \(5\).

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