Vector Algebra Question 90
Question: If a, b, c are coplanar vectors, then
[IIT 1989]
Options:
A) $ | ,\begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{b} & \mathbf{c} & \mathbf{a} \\ \mathbf{c} & \mathbf{a} & \mathbf{b} \\ \end{matrix}, |=\mathbf{0} $
B) $ | ,\begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a},.,\mathbf{a} & \mathbf{a},.,\mathbf{b} & \mathbf{a},.,\mathbf{c} \\ \mathbf{b},.,\mathbf{a} & \mathbf{b},.,\mathbf{b} & \mathbf{b},.,\mathbf{c} \\ \end{matrix}, |=\mathbf{0} $
C) $ | ,\begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{c},.,\mathbf{a} & \mathbf{c},.,\mathbf{b} & \mathbf{c},.,\mathbf{c} \\ \mathbf{b},.,\mathbf{a} & \mathbf{b},.,\mathbf{c} & \mathbf{b},.,\mathbf{b} \\ \end{matrix}, |=\mathbf{0} $
D) $ | ,\begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a},.,\mathbf{b} & \mathbf{a},.,\mathbf{a} & \mathbf{a},.,\mathbf{c} \\ \mathbf{c},.,\mathbf{a} & \mathbf{c},.,\mathbf{c} & \mathbf{c},.,\mathbf{b} \\ \end{matrix}, |=\mathbf{0} $
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Answer:
Correct Answer: B
Solution:
- Since $ \mathbf{a},\mathbf{b} $ and $ \mathbf{c} $ are coplanar, therefore there exists $ (x,,y,,z $ not all zero) such that $ x\mathbf{a}+y\mathbf{b}+z\mathbf{c}=0 $ …..(i) Multiply be $ \mathbf{a} $ scalarly, we get $ x(\mathbf{a},.,\mathbf{a})+(\mathbf{a},.,\mathbf{b})+z(\mathbf{a},.,\mathbf{c})=0 $ ……(ii) and $ x(\mathbf{a},.,\mathbf{b})+y(\mathbf{b},.,\mathbf{b})+z(\mathbf{b},.,\mathbf{c})=0 $ …..(iii) Eliminating $ x,,y $ and $ z $ from (i), (ii) and (iii), we get $ | ,\begin{matrix} \mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a},.,\mathbf{a} & \mathbf{a},.,\mathbf{b} & \mathbf{a},.,\mathbf{c} \\ \mathbf{a},.,\mathbf{b} & \mathbf{b},.,\mathbf{b} & \mathbf{b},.,\mathbf{c} \\ \end{matrix}, |=0 $ . Note: Students should remember this question as a formula.
BETA
