Question

The vectors $\vec{a}$ and $\vec{b}$ act in a plane as shown below. The magnitude of the vector} \[ \vec{c} = (\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) \] is

• A. zero.
• B. half to the area bounded by the vectors $\vec{a}$ and $\vec{b}$.
• C. equal to the area bounded by the vectors $\vec{a}$ and $\vec{b}$.
• D. twice the area bounded by the vectors $\vec{a}$ and $\vec{b}$.

Hint:

Always use $(\vec{b}\times\vec{a}) = -(\vec{a}\times\vec{b})$ to simplify expressions involving paired vectors.

Answer 1

The Correct Option is D

To evaluate the magnitude of \[ (\vec{a}+\vec{b}) \times (\vec{a}-\vec{b}), \] we expand the cross product term-by-term: \[ (\vec{a}+\vec{b}) \times (\vec{a}-\vec{b}) = \vec{a}\times\vec{a} - \vec{a}\times\vec{b} + \vec{b}\times\vec{a} - \vec{b}\times\vec{b}. \]

Step 1: Eliminate zero cross products.
Any vector crossed with itself is zero: \[ \vec{a}\times\vec{a} = 0, \vec{b}\times\vec{b} = 0. \] So the expression simplifies to: \[ -\vec{a}\times\vec{b} + \vec{b}\times\vec{a}. \]

Step 2: Use the anti-commutative property.
Cross product satisfies \[ \vec{b}\times\vec{a} = -(\vec{a}\times\vec{b}). \] Substituting this, \[ -\vec{a}\times\vec{b} - \vec{a}\times\vec{b} = -2(\vec{a}\times\vec{b}). \]

Step 3: Take magnitude.
\[ |\vec{c}| = |-2(\vec{a}\times\vec{b})| = 2|\vec{a}\times\vec{b}|. \] But \[ |\vec{a}\times\vec{b}| = \text{area of parallelogram formed by }\vec{a},\vec{b}. \] Thus \[ |\vec{c}| = \boxed{\text{twice the area}}. \]

Final Answer: Twice the area bounded by $\vec{a}$ and $\vec{b}$.