Question

Find the unit vector perpendicular to vectors Ā=2î+j+3k and B={+2j+ k.
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Answer 1

Given

$\rm \vec{A}=2i+j+3k\\\\\rm \vec B=i+2j+k$

To Find

unit vector perpendicular to the given

Solution

Vector perpendicular to A and B is " A × B "

$\bf \vec{A}\times \vec{B}\\\\\to \rm \left[\begin{array}{ccc}i&-j&k\\2&1&3\\1&2&1\end{array}\right] \\\\\to \rm i(1-6)-j(2-3)+k(4-1)\\\\\to \rm 5i+j+3k$

So ,

$\bf \pink{\bigstar\ \; \vec{A}\times \vec{B}=5i+j+3k}$

$\bf |\vec{A}\times \vec{B}|\\\\\to \rm \sqrt{5^2+1^2+3^2}\\\\\to \rm \sqrt{25+1+9}\\\\\to \bf \sqrt{35}$

Now , Unit vector perpendicular to A and B is ,

$\to \rm \dfrac{\vec{A}\times \vec{B}}{|\vec{A}\times \vec{B}|}\\\\\to \rm \dfrac{5i+j+3k}{\sqrt{35}}\\\\\to \bf \green{\dfrac{5}{\sqrt{35}}i+\dfrac{1}{\sqrt{35}}j+\dfrac{3}{\sqrt{35}}k\ \; \bigstar}$