If we know that $\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }$ , which of the following statements is true?
A)b and c are parallel
B) $| \mathbf { b } | = | \mathbf { c } |$
C)b = c
D)a is parallel to b - c

E)a is a unit vector

Question

If we know that $\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }$ , which of the following statements is true?
A)b and c are parallel
B) $| \mathbf { b } | = | \mathbf { c } |$ 
C)b = c
D)a is parallel to b - c

E)a is a unit vector

Quizplus · Accepted Answer

The cross product $\mathbf{a} \times \mathbf{b} = \mathbf{a} \times \mathbf{c}$ implies that $\mathbf{a}$ is perpendicular to the vector difference $\mathbf{b} - \mathbf{c}$. However, for vectors $\mathbf{a}$ and $\mathbf{d} = \mathbf{b} - \mathbf{c}$ to be parallel, it means that $\mathbf{a}$ must be perpendicular to any vector that is perpendicular to $\mathbf{d}$, which is not directly implied here. The correct interpretation is that $\mathbf{a}$ is orthogonal to $\mathbf{b} - \mathbf{c}$, not necessarily parallel in the conventional sense of having the same or opposite direction. The statement is misleading as written; a more accurate description would be that $\mathbf{a}$ is orthogonal to $\mathbf{b} - \mathbf{c}$, not parallel.

If We Know That a×b=a×c\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }a×b=a×c

If We Know That $\mathbf { a } \times \mathbf { b } = \mathbf { a } \times \mathbf { c }$