Question-33 If \vec{a}, \vec{b}, \vec{c} are three non – coplanar vector, then \frac{\vec{a} \cdot(\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}}+\frac{\vec{b} \cdot(\vec{a} \times \vec{c})}{\vec{c} \cdot(\vec{a} \times \vec{b})}=
A) 0
B) 2
C) -2
D) None
Answer: A) 0
Explanation
\begin{aligned} & \frac{\vec{a} \cdot(\vec{b} \times \vec{c})}{(\vec{c} \times \vec{a}) \cdot \vec{b}}+\frac{\vec{b} \cdot(\vec{a} \times \vec{c})}{\vec{c} \cdot(\vec{a} \times \vec{b})} \ & =\frac{\left[\begin{array}{lll} \vec{a} & \vec{b} & \vec{c} \end{array}\right]}{\left[\begin{array}{lll} \vec{c} & \vec{a} & \vec{b} \end{array}\right]}+\frac{\left[\begin{array}{lll} \vec{b} & \vec{a} & \vec{c} \end{array}\right]}{\left[\begin{array}{lll} \vec{c} & \vec{a} & \vec{b} \end{array}\right]} \ & =\frac{\left[\begin{array}{lll} \vec{a} & \vec{b} & \vec{c} \end{array}\right]}{\left[\begin{array}{lll} \vec{c} & \vec{a} & \vec{b} \end{array}\right]}-\frac{\left[\begin{array}{lll} \vec{a} & \vec{b} & \vec{c} \end{array}\right]}{\left[\begin{array}{lll} \vec{c} & \vec{a} & \vec{b} \end{array}\right]}=0 \end{aligned}