Question

If the magnitudes of two vectors are doubled, how will the magnitude of the cross product of the vectors change? Explain.

Short answer

After doubling the magnitudes of the vectors, the magnitude of the cross product becomes four times as large as it originally was.

Step-by-step solution

Understand the vector cross product property

The cross product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by \( \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin(\theta) \vec{n} \) where \( |\vec{A}| \) and \( |\vec{B}| \) are the magnitudes of the vectors, \( \theta \) is the angle between them, and \( \vec{n} \) is a unit vector perpendicular to the plane containing \( \vec{A} \) and \( \vec{B} \). The magnitude of the cross product is hence \( |\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin(\theta) \).

Apply scalar multiplication to each vector

If each vector's magnitude is doubled, this is equivalent to multiplying each vector by a scalar of 2. Our vectors now become \( 2\vec{A} \) and \( 2\vec{B} \).

Find the magnitude of the cross product of the doubled vectors

The cross product of our new vectors \( 2\vec{A} \) and \( 2\vec{B} \) is given by \( 2\vec{A} \times 2\vec{B} = 4 |\vec{A}| |\vec{B}| \sin(\theta) \vec{n} \). Hence, the magnitude of the cross product of the doubled vectors is \( 4 |\vec{A}| |\vec{B}| \sin(\theta) \). This is four times the original magnitude of the cross product.