Question

If $\overline{x},\overline{y},\overline{z}$ are mutually perpendicular unit vectors and $a\hat{x}+a\hat{y}+c\hat{z}$, $c\hat{x}+c\hat{y}+b\hat{z}$ are coplanar, then

• A. $a,b,c$ are in A.P
• B. $a,b,c$ are in G.P
• C. $a,b,c$ are in H.P
• D. $a,c,b$ are in G.P

Answer 1

The correct option is D $a,c,b$ are in G.P

As $\overrightarrow{x},\overrightarrow{y},\overrightarrow{z}$ are mutually perpendicular, hence they are linearly independent, hence ratio of coefficients linking them are equal.

$\Rightarrow \frac{a}{c}=\frac{a}{c}=\frac{c}{b}$

$\Rightarrow \frac{a}{c}=\frac{c}{b}$

$\Rightarrow c^2=ab$

$\Rightarrow a,c,b$ are in G.P

$\Rightarrow \left(D\right)$.