Question

The altitude through vertex A of $\triangle ABC$ with position vectors of points A, B, C as $\vec{a}, \vec{b}, \vec{c}$ respectively is ______.

• A. $|\vec{b} \times \vec{c}| / |\vec{c} - \vec{b}|$
• B. $|\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}| / |\vec{c} - \vec{b}|$
• C. $|\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}| / |\vec{c} \times \vec{b}|$
• D. $|\vec{b} \times \vec{c}| / |\vec{a}|$

Hint:

The vector quantity $(\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a})$ is incredibly important. Its magnitude directly represents exactly twice the area of the triangle formed by the three position vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The area of a triangle can be expressed as $\frac{1}{2} \times \text{Base} \times \text{Altitude}$. Therefore, $\text{Altitude} = \frac{2 \times \text{Area}}{\text{Base}}$.

Step 2: Formula Application:
Area of $\triangle ABC = \frac{1}{2} |(\vec{b}-\vec{a}) \times (\vec{c}-\vec{a})| = \frac{1}{2} |\vec{b} \times \vec{c} - \vec{b} \times \vec{a} - \vec{a} \times \vec{c} + \vec{a} \times \vec{a}|$. Since $\vec{a} \times \vec{a} = 0$, Area $= \frac{1}{2} |\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|$.

Step 3: Explanation:
The altitude is through vertex A, so the base is the side $BC$. Base length $= |\vec{c} - \vec{b}|$. Altitude $= \frac{2 \times \frac{1}{2} |\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|}{|\vec{c} - \vec{b}|}$.

Step 4: Final Answer:
The altitude is $\frac{|\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|}{|\vec{c} - \vec{b}|}$.