IIT-JEE 2012 · Paper 2, Mathematics question 43

Rearrange. With $\vec c=2\hat i+3\hat j+4\hat k$: $\vec a\times\vec c=\vec c\times\vec b=-\vec b\times\vec c$, so
$$(\vec a+\vec b)\times\vec c=\vec0\implies\vec a+\vec b=\lambda\vec c.$$

Magnitude. $|\vec c|=\sqrt{29}$, so $\lambda=\pm1$.

Dot product. $\vec c\cdot(-7\hat i+2\hat j+3\hat k)=-14+6+12=4$, so the value is $\pm4$. A possible value is 4.

(C)