Calculate the angle between two vectors having magnitudes 3 and 4. The magnitude of their dot product is `2sqrt2`.

Answer

To calculate the angle between two vectors having magnitudes 3 and 4, and a dot product of \(2\sqrt{2}\), we can follow these steps:

### Step 1: Understand the dot product formula
The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is given by the formula:
\[ \mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta \]
where \( |\mathbf{A}| \) and \( |\mathbf{B}| \) are the magnitudes of the vectors, and \( \theta \) is the angle between them.

### Step 2: Substitute the known values
We know:
- \( |\mathbf{A}| = 3 \)
- \( |\mathbf{B}| = 4 \)
- \( \mathbf{A} \cdot \mathbf{B} = 2\sqrt{2} \)

Substituting these values into the dot product formula, we get:
\[ 2\sqrt{2} = 3 \cdot 4 \cdot \cos \theta \]

### Step 3: Simplify the equation
Now, simplify the equation:
\[ 2\sqrt{2} = 12 \cos \theta \]

### Step 4: Solve for \( \cos \theta \)
To find \( \cos \theta \), divide both sides by 12:
\[ \cos \theta = \frac{2\sqrt{2}}{12} \]
\[ \cos \theta = \frac{\sqrt{2}}{6} \]

### Step 5: Calculate \( \theta \)
Now, to find the angle \( \theta \), take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{\sqrt{2}}{6}\right) \]

### Step 6: Use a calculator to find the angle
Using a scientific calculator, we can find:
\[ \theta \approx 76.35^\circ \]

### Final Answer
The angle between the two vectors is approximately \( 76.35^\circ \).

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