Given the vectors a and b orthogonal to each other find the vector V in terms of a and b where V.a=0 ,V.b=1 and [V a b]=1 ??

Question

Given the vectors a and b orthogonal to each other find the vector V in terms of a and b where V.a=0 ,V.b=1 and [V a b]=1 ?? I am not getting any way to solve this cud u pls help me with this and pls give proper and full explanation.

Answer 1

Hope I understood properly.

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We have $\vec a. \vec v =0$. Also $\vec a\times \vec b$ is perpendicular to both $\vec a, \vec b$ and $\vec a\times \vec b$ are mutually perpendicular vectors. We also know that $[\vec a\vec a\vec b] = [\vec b \vec a\vec b] = 0$.

Therefore $$\vec v = p\vec a+q\vec b+r(\vec a\times \vec b)...(1)$$ Now, as $\vec v. \vec a=0 \Rightarrow p=0$. Also, $\vec v.\vec b=1\Rightarrow 0+q|b^2|+0=1 \Rightarrow q=\frac{1}{|b^2|}$.

We have $\vec v.(\vec a\times \vec b) = p.0 + q.0 + r|\vec a\times \vec b|^2 =1 \Rightarrow [\vec v \vec a \vec b] = r|\vec a\times \vec b|^2 = 1\Rightarrow r= \frac{1}{|\vec a\times \vec b|^2}$.

Now substitute in $(1)$. Hope it helps.