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Vector or Cross Product — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSVector or Cross Product3 Marks
Question
If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are three unit vectors such that $\vec{\text{a}}\times\vec{\text{b}}=\vec{\text{c}},\vec{\text{b}}\times\vec{\text{c}}=\vec{\text{a}},\vec{\text{c}}\times\vec{\text{a}}=\vec{\text{b}}.$Show that $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ from an orthonormal right handed triad of unit vectors.
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Answer

Given:
$\vec{\text{a}}\times\vec{\text{b}}=\vec{\text{c}}$
$\vec{\text{b}}\times\vec{\text{c}}=\vec{\text{a}}$
$\vec{\text{c}}\times\vec{\text{a}}=\vec{\text{b}}\dots(1)$
Now,
$\big|\vec{\text{a}}\times\vec{\text{b}}\big|=|\vec{\text{c}}|=1$ ( $\because\vec{\text{c}}$ is a unit vector)
$\big|\vec{\text{b}}\times\vec{\text{c}}\big|=|\vec{\text{a}}|=1$ ($\because\vec{\text{a}}$ is a unit vector)
$\big|\vec{\text{c}}\times\vec{\text{a}}\big|=\big|\vec{\text{b}}\big|=1$ ($\because\vec{\text{b}}$ is a unit vector)
$\therefore\big|\vec{\text{a}}\times\vec{\text{b}}\big|=\big|\vec{\text{b}}\times\vec{\text{c}}\big|=\big|\vec{\text{c}}\times\vec{\text{a}}\big|=1\dots(2)$
From (1) and (2), we know
$\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ form an orthonormal right handed triad of unit vectors.

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