Question

If the vectors $c$, $a=xi+yj+zk$ and $b=j$ are such that $a$, $c$ and $b$ form a right handed system. Then $c$ is?
(1) $zi-xk$
(2) $0$
(3) $yj$
(4) $-zi+xk$

Answer 1

In this type of question we have to use the concept of vectors. We know that the three vectors $u,v,w$ form a right hand system if when we extend the fingers of our right hand along the direction of vector $u$ and curl them in the direction of $v$ then the thumb points roughly in the direction of $w$. Hence we can say that if three vectors $u={{u}_{1}}i+{{u}_{2}}j+{{u}_{3}}k$, $v={{v}_{1}}i+{{v}_{2}}j+{{v}_{3}}k$ and $w={{w}_{1}}i+{{w}_{2}}j+{{w}_{3}}k$ form a right handed system then $w=u\times v$. Also we know that the cross product of two vectors say $u={{u}_{1}}i+{{u}_{2}}j+{{u}_{3}}k$ and $v={{v}_{1}}i+{{v}_{2}}j+{{v}_{3}}k$ is defined as $u\times v=\left| \begin{matrix} i & j & k \\\ {{u}_{1}} & {{u}_{2}} & {{u}_{3}} \\\ {{v}_{1}} & {{v}_{2}} & {{v}_{3}} \\\ \end{matrix} \right|$

Complete step-by-step solution:
Now we have to find the vector $c$ such that the vectors $a$, $c$ and $b$ form a right handed system where $a=xi+yj+zk$ and $b=j$
Now we know that if the three vectors $u,v,w$ form a right handed system then $w=u\times v$so as the vectors $a$, $c$ and $b$ form a right handed system we can write
$\Rightarrow c=b\times a$
Now by considering the cross product of $a$ and $b$ we get

i & j & k \\\ 0 & 1 & 0 \\\ x & y & z \\\ \end{matrix} \right|$$ On simplification we get, $$\begin{aligned} & \Rightarrow c=i\left[ z-0 \right]-j\left[ 0-0 \right]+k\left[ 0-x \right] \\\ & \Rightarrow c=zi-xk \\\ \end{aligned}$$ **Hence, option (1) is the correct option.** **Note:** In this type of question students have to take care in calculation of cross product. Students have to note that depending on the order of cross product answers get changed in sign so that they have to remember to take the cross product in anti-clockwise direction.