For any two vectors \(\vec{a}\) and \(\vec{b}\), \((2\vec{a}+\vec{b})\times(\vec{a}-3\vec{b})\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(-7(\vec{a}\times\vec{b})\)
Expanding by the distributive law:
\((2\vec{a}+\vec{b})\times(\vec{a}-3\vec{b}) = 2(\vec{a}\times\vec{a})-6(\vec{a}\times\vec{b})+(\vec{b}\times\vec{a})-3(\vec{b}\times\vec{b})\)
A vector crossed with itself is zero, and \(\vec{b}\times\vec{a} = -(\vec{a}\times\vec{b})\) (the cross product is not commutative):
\( = 0-6(\vec{a}\times\vec{b})-(\vec{a}\times\vec{b})-0\)
\( = -7(\vec{a}\times\vec{b})\)
Why the other options are incorrect
Option (1) \(7(\vec{a}\times\vec{b})\) — Incorrect. Both remaining terms are negative, \(-6(\vec{a}\times\vec{b})\) and \(-(\vec{a}\times\vec{b})\), so the sign must be minus.
Option (2) \(-5(\vec{a}\times\vec{b})\) — Incorrect. It takes \(\vec{b}\times\vec{a} = +(\vec{a}\times\vec{b})\), giving \(-6+1\); the order of a cross product cannot be swapped without a sign change.
Option (3) \(5(\vec{a}\times\vec{b})\) — Incorrect. Neither the sign nor the number 5 matches \(-6-1 = -7\).