Vector Algebra
Cross Product
Grade 12
Question:
<p>Let \(\vec{a},\vec{b}\) be two non-zero non-collinear vectors. Then \(|\vec{a}\times\vec{b}|\) equals</p>
<li>\(|\vec{a}||\vec{b}|\sin\theta\)</li>
<li>\(|\vec{a}||\vec{b}|\cos\theta\)</li>
<li>\(|\vec{a}||\vec{b}|\)</li>
<li>\(|\vec{a}||\vec{b}|\sin\theta\) with \(\vec{a}\times\vec{b}\perp\vec{a}\) and \(\vec{a}\times\vec{b}\perp\vec{b}\)</li>
Step-by-Step Solution
Key Concept: Cross product magnitude is |a||b|sin\theta AND the result is perpendicular to both — all of these properties together define the cross product.
By definition, $\vec{a}\times\vec{b}$ is a vector with:
<ul><li>Magnitude $|\vec{a}||\vec{b}|\sin\theta$</li>
<li>Direction perpendicular to both $\vec{a}$ and $\vec{b}$</li></ul>
Option D states both the magnitude formula AND the perpendicularity — making it the most complete and correct statement. Answer: (D)
Correct Answer: D