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CSS Applied Mathematics QUESTION #1294
Question 1
If \(\vec{a}\times(\vec{b}\times\vec{c}) = (\vec{a}\times\vec{b})\times\vec{c}\), what can be concluded about \(\vec{a}\) and \(\vec{c}\)?
  • They are perpendicular
  • They are parallel✔️
  • Their magnitudes are equal
  • Their dot product is 1
Correct Answer Explanation
Expand both sides using the BAC-CAB rule. LHS: \(\vec{b}(\vec{a}\cdot\vec{c})-\vec{c}(\vec{a}\cdot\vec{b})\). RHS: \(\vec{a}(\vec{b}\cdot\vec{c})-\vec{b}(\vec{a}\cdot\vec{c})\). Setting equal and simplifying yields \((\vec{a}\times\vec{c})\times\vec{b}=\vec{0}\) for all \(\vec{b}\), which implies \(\vec{a}\times\vec{c}=\vec{0}\), meaning \(\vec{a}\) and \(\vec{c}\) are parallel.