Vector Algebra
Cross Product
Grade 12
Question:
<p>For two particular vectors <strong>A</strong> and <strong>B</strong>, it is known that <strong>A</strong> × <strong>B</strong> = <strong>B</strong> × <strong>A</strong>. What must be true about the two vectors?</p>
<p>(a) Atleast one of the two vectors must be the zero vector</p>
<p>(b) <strong>A</strong> × <strong>B</strong> = <strong>B</strong> × <strong>A</strong> is true for any two vectors</p>
<p>(c) One of the two vectors is a scalar multiple of the other vector</p>
<p>(d) The two vectors must be perpendicular to each other</p>
Step-by-Step Solution
Key Concept: Use the anti-commutative property of the cross product to determine when equality holds.
A × B = − B × A by the property of cross product. If A × B = B × A , then A × B = − A × B , which implies 2( A × B ) = 0 , so A × B = 0 . This means the vectors are parallel, i.e., one is a scalar multiple of the other, or at least one is zero. The simplest case is that one vector is zero.
Correct Answer: A