If `|a|=10, |b|=2 and a.b=12,` then value of `|a xx b|` is

Options

A. 5
B. 10
C. 14
D. 16

Explanation

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

### Step-by-Step Solution:

1. **Given Information:**
- \(|a| = 10\)
- \(|b| = 2\)
- \(a \cdot b = 12\)

2. **Using the Dot Product Formula:**
The dot product of two vectors can be expressed as:
\[ a \cdot b = |a| |b| \cos \theta \]
where \(\theta\) is the angle between the vectors \(a\) and \(b\).

3. **Substituting the Given Values:**
Substitute the known values into the dot product formula:
\[ 12 = 10 \cdot 2 \cdot \cos \theta \]
Simplifying this gives:
\[ 12 = 20 \cos \theta \]

4. **Solving for \(\cos \theta\):**
Rearranging the equation to find \(\cos \theta\):
\[ \cos \theta = \frac{12}{20} = \frac{3}{5} \]

5. **Finding \(\sin \theta\):**
We can use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \(\cos \theta\):
\[ \sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1 \]
\[ \sin^2 \theta + \frac{9}{25} = 1 \]
\[ \sin^2 \theta = 1 - \frac{9}{25} = \frac{25 - 9}{25} = \frac{16}{25} \]
\[ \sin \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \]

6. **Finding \(|a \times b|\):**
The magnitude of the cross product is given by:
\[ |a \times b| = |a| |b| \sin \theta \]
Substituting the values we have:
\[ |a \times b| = 10 \cdot 2 \cdot \frac{4}{5} \]
\[ |a \times b| = 20 \cdot \frac{4}{5} = 16 \]

### Final Answer:
The value of \(|a \times b|\) is \(16\).

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