If `|A|=2,|B|=5` and `|AxxB|=8.` Angle between A and B is acute, then `(A.B)` is

Options

A. 6
B. 3
C. 4
D. 7

Explanation

To solve the problem step by step, we will use the properties of vectors, specifically the relationships between the dot product and cross product.

### Step 1: Write down the given information
We are given:
- Magnitude of vector A, |A| = 2
- Magnitude of vector B, |B| = 5
- Magnitude of the cross product, |A × B| = 8
- The angle between vectors A and B is acute.

### Step 2: Use the formula for the magnitude of the cross product
The magnitude of the cross product of two vectors A and B is given by:
\[ |A \times B| = |A| |B| \sin \theta \]
Substituting the known values:
\[ 8 = 2 \cdot 5 \cdot \sin \theta \]

### Step 3: Simplify the equation
Now simplify the equation:
\[ 8 = 10 \sin \theta \]
To find sin θ, divide both sides by 10:
\[ \sin \theta = \frac{8}{10} = \frac{4}{5} \]

### Step 4: Use the Pythagorean identity to find cos θ
Using the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting sin θ:
\[ \left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1 \]
Calculating:
\[ \frac{16}{25} + \cos^2 \theta = 1 \]
\[ \cos^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
Taking the square root (since the angle is acute, cos θ is positive):
\[ \cos \theta = \frac{3}{5} \]

### Step 5: Use the formula for the dot product
The dot product of two vectors A and B is given by:
\[ A \cdot B = |A| |B| \cos \theta \]
Substituting the known values:
\[ A \cdot B = 2 \cdot 5 \cdot \frac{3}{5} \]

### Step 6: Simplify to find A · B
Calculating:
\[ A \cdot B = 2 \cdot 3 = 6 \]

### Final Answer
Thus, the value of \( A \cdot B \) is **6**.

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