Vector Algebra Question 34
Question: If the vectors $ \alpha \hat{i}+\alpha \hat{j}+\gamma \hat{k},\hat{i}+\hat{k} $ and $ \gamma \hat{i}+\gamma \hat{j}+\beta \hat{k} $ lie on a plane, where $ \alpha ,\beta $ and $ \gamma $ are distinct non-negative numbers, then $ \gamma $ is
Options:
A) Arithmetic mean of $ \alpha $ and $ \beta $
B) Geometric mean of $ \alpha $ and $ \beta $
C) Harmonic mean of $ \alpha $ and $ \beta $
D) None of the above
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] If three vectors are co-planar.
$ \Rightarrow \begin{vmatrix} \alpha & \alpha & \gamma \\ 1 & 0 & 1 \\ \gamma & \gamma & \beta \\ \end{vmatrix} =0 $
$ \Rightarrow \alpha [0-\gamma ]-\alpha [\beta +\gamma ]+\gamma [\gamma -0]=0 $
$ \Rightarrow -\alpha \gamma -\alpha \beta +\alpha \gamma +{{\gamma }^{2}}=0 $
$ \Rightarrow {{\gamma }^{2}}=\alpha \beta $
$ \Rightarrow $ So $ \alpha ,\beta ,\gamma $ are in G.P.
BETA
