Question

$\vec{a}, \vec{b}, \vec{c}, \vec{d}$ be non-zero vectors such that $\vec{a}\times\vec{b}=\vec{c}\times\vec{d}$ and $\vec{a}\times\vec{c}=\vec{b}\times\vec{d}$. Then:

• A. $\vec{a}-\vec{d}$ is parallel to $\vec{b}-\vec{c}$
• B. $\vec{a}-\vec{b}$ is parallel to $\vec{b}-\vec{c}$
• C. $\vec{b}-\vec{c}$ is parallel to $\vec{b}+\vec{c}$
• D. $\vec{a}-\vec{c}$ is parallel to $\vec{b}-\vec{c}$
• E. $\vec{a}+\vec{c}$ is parallel to $\vec{b}+\vec{d}$

Hint:

The cross product $\vec{u} \times \vec{v} = 0$ implies vectors $\vec{u}$ and $\vec{v}$ are collinear or parallel.

Answer 1

The Correct Option is A

Step 1: Concept
Subtracting the two given cross product equations helps identify common factors.

Step 2: Analysis
Subtracting $\vec{a}\times\vec{c} = \vec{b}\times\vec{d}$ from $\vec{a}\times\vec{b} = \vec{c}\times\vec{d}$: $(\vec{a}\times\vec{b}) - (\vec{a}\times\vec{c}) = (\vec{c}\times\vec{d}) - (\vec{b}\times\vec{d})$ $\vec{a} \times (\vec{b} - \vec{c}) = (\vec{c} - \vec{b}) \times \vec{d}$ $\vec{a} \times (\vec{b} - \vec{c}) + \vec{d} \times (\vec{b} - \vec{c}) = 0$ $(\vec{a} - \vec{d}) \times (\vec{b} - \vec{c}) = 0$

Step 3: Conclusion
If the cross product of two non-zero vectors is zero, the vectors are parallel. Final Answer: (A)