If `abs(a)=2, abs(b)=5" and "abs(a times b)=8" then "abs(a*b)` is equal to

Options

A. 4
B. 6
C. 5
D. none of these

Answer

To solve the problem, we need to find the value of \( |a \cdot b| \) given the magnitudes of vectors \( a \) and \( b \), and the magnitude of their cross product.

### Step-by-step solution:

1. **Given Information**:
- \( |a| = 2 \)
- \( |b| = 5 \)
- \( |a \times b| = 8 \)

2. **Using the Formula for Cross Product**:
The magnitude of the cross product of two vectors is given by:
\[ |a \times b| = |a| |b| \sin \theta \]
where \( \theta \) is the angle between the vectors \( a \) and \( b \).

3. **Substituting the Known Values**:
Substitute the known values into the formula:
\[ 8 = 2 \cdot 5 \cdot \sin \theta \]
Simplifying this gives:
\[ 8 = 10 \sin \theta \]
Therefore, we can solve for \( \sin \theta \):
\[ \sin \theta = \frac{8}{10} = \frac{4}{5} \]

4. **Using the Pythagorean Identity**:
We know from trigonometry that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \sin \theta \):
\[ \left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1 \]
This simplifies to:
\[ \frac{16}{25} + \cos^2 \theta = 1 \]
Therefore:
\[ \cos^2 \theta = 1 - \frac{16}{25} = \frac{9}{25} \]
Taking the positive square root (since cosine can be positive):
\[ \cos \theta = \frac{3}{5} \]

5. **Finding the Dot Product**:
The magnitude of the dot product of two vectors is given by:
\[ |a \cdot b| = |a| |b| \cos \theta \]
Substituting the known values:
\[ |a \cdot b| = 2 \cdot 5 \cdot \frac{3}{5} \]
This simplifies to:
\[ |a \cdot b| = 2 \cdot 3 = 6 \]

### Final Answer:
Thus, \( |a \cdot b| = 6 \).

---