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

Options

A. 6
B. 7
C. 8
D. 9

Answer

To solve the problem step by step, we will use the given information about the magnitudes of vectors \( \mathbf{a} \) and \( \mathbf{b} \), as well as the magnitude of their cross product.

### Given:
- \( |\mathbf{a}| = 2 \)
- \( |\mathbf{b}| = 5 \)
- \( |\mathbf{a} \times \mathbf{b}| = 8 \)

### Step 1: Use the formula for the magnitude of the cross product
The magnitude of the cross product of two vectors is given by the formula:
\[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin \theta \]
where \( \theta \) is the angle between the vectors \( \mathbf{a} \) and \( \mathbf{b} \).

### Step 2: Substitute the known values into the formula
Substituting the known values into the formula, we have:
\[ 8 = 2 \times 5 \times \sin \theta \]
This simplifies to:
\[ 8 = 10 \sin \theta \]

### Step 3: Solve for \( \sin \theta \)
To find \( \sin \theta \), we can rearrange the equation:
\[ \sin \theta = \frac{8}{10} = \frac{4}{5} \]

### Step 4: Use the Pythagorean identity to find \( \cos \theta \)
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \sin \theta = \frac{4}{5} \) into the equation:
\[ \left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1 \]
This gives:
\[ \frac{16}{25} + \cos^2 \theta = 1 \]
Rearranging this, we find:
\[ \cos^2 \theta = 1 - \frac{16}{25} = \frac{25 - 16}{25} = \frac{9}{25} \]
Taking the square root gives:
\[ \cos \theta = \frac{3}{5} \]

### Step 5: Calculate \( \mathbf{a} \cdot \mathbf{b} \)
The dot product of two vectors is given by:
\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \]
Substituting the known values:
\[ \mathbf{a} \cdot \mathbf{b} = 2 \times 5 \times \frac{3}{5} \]
The \( 5 \) cancels out:
\[ \mathbf{a} \cdot \mathbf{b} = 2 \times 3 = 6 \]

### Final Answer:
Thus, the value of \( \mathbf{a} \cdot \mathbf{b} \) is \( 6 \).

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