To calculate the magnitude of the vector cross product $|\vec{a} \times \vec{b}|$, we use the formula:

$|\vec{a} \times \vec{b}| = |\vec{a}| \cdot |\vec{b}| \cdot \sin\theta$

Here:

• $|\vec{a}|$ is the magnitude of vector $\vec{a}$,
• $|\vec{b}|$ is the magnitude of vector $\vec{b}$,
• $\theta$ is the angle between the vectors.

From the given data: $|\vec{a}| = 3, \quad |\vec{b}| = 20, \quad \cos\theta = 30$

We calculate $\sin\theta$ using the Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$ $\sin\theta = \sqrt{1 - \cos^2\theta}$

Substituting $\cos\theta = 30$: $\sin\theta = \sqrt{1 - (30)^2}$

However, this value for $\cos\theta$ appears too large for a valid cosine (since $|\cos\theta| \leq 1$). Could you clarify the problem or confirm the given value?

Let me know,