If `|a|= 2, |b|=5 and |a xx b|=8`, then |a.b| is equal to

Options

A. 3
B. 4
C. 5
D. 6

Answer

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

Given:
- |a| = 2
- |b| = 5
- |a × b| = 8

We need to find |a · b|.

### Step 1: 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:
\[ |a \times b| = |a| |b| \sin \theta \]
where \(\theta\) is the angle between the vectors a and b.

### Step 2: Substitute the known values into the formula
Substituting the given values into the formula:
\[ 8 = (2)(5) \sin \theta \]
This simplifies to:
\[ 8 = 10 \sin \theta \]

### Step 3: Solve for \(\sin \theta\)
Now, we can solve for \(\sin \theta\):
\[ \sin \theta = \frac{8}{10} = \frac{4}{5} \]

### Step 4: Use the relationship between sine and cosine
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \(\sin \theta\) into this equation:
\[ \left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1 \]
This gives:
\[ \frac{16}{25} + \cos^2 \theta = 1 \]
\[ \cos^2 \theta = 1 - \frac{16}{25} = \frac{9}{25} \]
Taking the square root:
\[ \cos \theta = \frac{3}{5} \quad (\text{since cosine is positive in the range of angles considered}) \]

### Step 5: Use the formula for the dot product
The dot product of two vectors is given by:
\[ |a \cdot b| = |a| |b| \cos \theta \]
Substituting the known values:
\[ |a \cdot b| = (2)(5) \left(\frac{3}{5}\right) \]
This simplifies to:
\[ |a \cdot b| = 10 \cdot \frac{3}{5} = 6 \]

### Final Answer
Thus, the magnitude of \(a \cdot b\) is:
\[ |a \cdot b| = 6 \]

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