If two vectors have nonzero magnitudes and their scalar product is zero

If two vectors have nonzero magnitudes and their scalar product is zero, it follows that

A) Their cross product is zero.

B) Their cross product has magnitude equal to the product of their magnitudes.

C) The vectors are parallel.

D) The vectors point in the opposite directions.

E) Their cross product has magnitude equal to 1.

The correct answer and explanation is:

The correct answer is B) Their cross product has magnitude equal to the product of their magnitudes.

Explanation:

When two vectors have a scalar (dot) product of zero, this means they are perpendicular to each other. The scalar product of two vectors A⃗\vec{A} and B⃗\vec{B} is given by: A⃗⋅B⃗=∣A⃗∣∣B⃗∣cos⁡θ\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta

Where θ\theta is the angle between the vectors, and ∣A⃗∣|\vec{A}| and ∣B⃗∣|\vec{B}| are the magnitudes of the vectors. If the scalar product is zero, cos⁡θ=0\cos \theta = 0, which means that the angle θ\theta is 90° (or π2\frac{\pi}{2} radians), indicating the vectors are perpendicular.

Now, the cross product of two vectors A⃗\vec{A} and B⃗\vec{B} is given by: A⃗×B⃗=∣A⃗∣∣B⃗∣sin⁡θ n^\vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin \theta \, \hat{n}

Where n^\hat{n} is the unit vector perpendicular to the plane formed by A⃗\vec{A} and B⃗\vec{B}. Since the vectors are perpendicular (θ=90∘\theta = 90^\circ), sin⁡90∘=1\sin 90^\circ = 1. Therefore, the magnitude of the cross product is: ∣A⃗×B⃗∣=∣A⃗∣∣B⃗∣|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}|

This confirms that the magnitude of the cross product is equal to the product of the magnitudes of the two vectors.

Other options:

• A) Their cross product is zero: This is false because the cross product is nonzero when the vectors are perpendicular.
• C) The vectors are parallel: This is false; if the scalar product is zero, the vectors are perpendicular, not parallel.
• D) The vectors point in opposite directions: This is false because the vectors are perpendicular, not anti-parallel.
• E) Their cross product has magnitude equal to 1: This is incorrect because the magnitude of the cross product depends on the magnitudes of the vectors, not just being equal to 1.

Therefore, the correct answer is B.