Vector Algebra
Cross Product and Area
Grade 12
Question:
<p>Unit vector perpendicular to the plane of \(\triangle ABC\) with position vectors \(\vec{a}, \vec{b}, \vec{c}\) of the vertices \(A, B, C\) is</p>
<p>(a) \(\frac{\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}}{\Delta}\)</p>
<p>(b) \(\frac{\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}}{2\Delta}\)</p>
<p>(c) \(\frac{\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}}{4\Delta}\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: The vector perpendicular to a plane is given by the cross product of two vectors in that plane. The area of the triangle is half the magnitude of this cross product.
Solution: \(\vec{AB} \times \vec{AC} = (\vec{b} - \vec{a}) \times (\vec{c} - \vec{a})\) \(= \vec{b} \times \vec{c} - \vec{b} \times \vec{a} - \vec{a} \times \vec{c} + \vec{a} \times \vec{a}\) \(= \vec{b} \times \vec{c} + \vec{a} \times \vec{b} + \vec{c} \times \vec{a}\) [since \(\vec{a} \times \vec{a} = 0\)] Area of \(\triangle ABC = \frac{1}{2}|\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|\) Therefore, \(\hat{n} = \frac{\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}}{|\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|}\) ∴ Answer is (b).
Correct Answer: B