Which statement correctly justifies that u×v is perpendicular to both u and v?

Which statement correctly justifies that u×v is perpendicular to both u and v?

• A. (u×v)·u = 0 and (u×v)·v = 0
• B. (u×v)×u = 0
• C. |u×v|^2 = |u|^2|v|^2 − (u·v)^2
• D. u×v is parallel to u

Answer: (A)

The cross product is a vector perpendicular to both original vectors. This can be seen through the dot product: if a vector is perpendicular to another, their dot product is zero. Since u×v is at right angles to u, the dot product (u×v)·u equals 0. Likewise, (u×v)·v also equals 0 because u×v is perpendicular to v. Therefore u×v is perpendicular to both u and v, which is exactly what the statement asserts. The other options miss this direct perpendicularity check or describe different relationships that don’t guarantee orthogonality.

The cross product is a vector perpendicular to both original vectors. This can be seen through the dot product: if a vector is perpendicular to another, their dot product is zero. Since u×v is at right angles to u, the dot product (u×v)·u equals 0. Likewise, (u×v)·v also equals 0 because u×v is perpendicular to v. Therefore u×v is perpendicular to both u and v, which is exactly what the statement asserts. The other options miss this direct perpendicularity check or describe different relationships that don’t guarantee orthogonality.