If `|veca|=10,|vecb|=2andveca.vecb=12`, then the value of `|vecaxxvecb|` is

Options

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

Explanation

To solve the problem, we need to find the magnitude of the cross product of two vectors \(\vec{a}\) and \(\vec{b}\) given the following information:

- \(|\vec{a}| = 10\)
- \(|\vec{b}| = 2\)
- \(\vec{a} \cdot \vec{b} = 12\)

The formula for the dot product of two vectors is given by:

\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]

Where \(\theta\) is the angle between the two vectors. From the information provided, we can substitute the known values into this formula:

\[ 12 = 10 \cdot 2 \cdot \cos \theta \]

Calculating the right side:

\[ 12 = 20 \cos \theta \]

Now, we can solve for \(\cos \theta\):

\[ \cos \theta = \frac{12}{20} = \frac{3}{5} \]

Next, we need to find \(\sin \theta\). We can use the Pythagorean identity:

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

Substituting \(\cos^2 \theta\):

\[ \sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1 \]

Calculating \(\left(\frac{3}{5}\right)^2\):

\[ \sin^2 \theta + \frac{9}{25} = 1 \]

Subtracting \(\frac{9}{25}\) from both sides:

\[ \sin^2 \theta = 1 - \frac{9}{25} = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} \]

Taking the square root of both sides, we find:

\[ \sin \theta = \pm \frac{4}{5} \]

Now, we can find the magnitude of the cross product \(|\vec{a} \times \vec{b}|\) using the formula:

\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]

Substituting the known values:

\[ |\vec{a} \times \vec{b}| = 10 \cdot 2 \cdot \left|\sin \theta\right| = 10 \cdot 2 \cdot \frac{4}{5} \]

Calculating this:

\[ |\vec{a} \times \vec{b}| = 20 \cdot \frac{4}{5} = 16 \]

Thus, the value of \(|\vec{a} \times \vec{b}|\) is:

\[ \boxed{16} \]