Let a= i-2 j+3 k, b= i+ j+ k, and c satisfy a+( b c)= 0 and b c=5.

Question

Let $\vec a=\hat i-2\hat j+3\hat k$, $\vec b=\hat i+\hat j+\hat k$, and $\vec c$ satisfy $\vec a+(\vec b	imes\vec c)=\vec 0$ and $\vec b\cdot\vec c=5$. Then $3\vec c\cdot\vec a$ equals\_\_\_\_

Answer: 0

Accepted Answer

*(The source explicitly flags this question as having a data inconsistency: solving $\vec b	imes\vec c=-\vec a$ together with $\vec b\cdot\vec c=5$ gives $\vec c=\dfrac13(10\hat i+3\hat j+2\hat k)$, from which $3\vec a\cdot\vec c=10$ --- but this does not satisfy $\vec a+\vec b	imes\vec c=\vec 0$ exactly, and moreover $\vec a\cdot\vec b=1-2+3=2
e0$, which contradicts the requirement $\vec a\perp\vec b$ implied by $\vec a=-\vec b	imes\vec c$. The official answer key gives 0.)*

Answer: 0

Solutions · 1