Given P=2i-3j, Q=i+j-k, R=3i-k, the vector V, which is orthogonal to P and Q and having a unit scalar product with R?

Problem – Given P=2i-3j, Q=i+j-k, R=3i-k, the vector V, which is orthogonal to P and Q and having a unit scalar product with R?

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Solution 

A unit vector along $\vec{V}$, which is orthogonal to both P and Q can be calculated as:

$\hat{V} = \frac{\vec{P}x\vec{Q}}{\vec{|P}x\vec{Q}|} = \frac{3i+2j+5k}{\sqrt{38}}$

Now, according to problem,

$\vec{V}.\vec{R}$ = 1 (Given)

so,

$\Longrightarrow$ (V$\hat{V}$).$\vec{R}$ = 1

$\Longrightarrow$ V = $\frac{\sqrt{38}}{4}$

Hence,

$\vec{V}$ = V$\hat{V}$ = $\frac{3i+2j+5k}{4}$