Why modulus is not considered in this vector algebra problem.

Question

Question: given two vectors $i-j$ and $i+2j$ , the unit vector coplanar with the two vectors and perpendicular to first is

(a)$\dfrac{1}{\sqrt 2} (i+j)$

(b)$\pm\dfrac{1}{\sqrt 2} (i +j)$

Let given two vectors be a and b, required vector be $r=xi+yj+zk$ then $[a b c]=0$ solving I got z=0. For solving r and a coplanar gives $x=y$.

Now $r=xi +xi$. Then unit vector means $|r|=\sqrt{x^2+x^2}=|x|\sqrt2=\pm x\sqrt2$

Unit vector $r=\dfrac{x(i +j)}{\pm x\sqrt2}=\pm\dfrac{1}{\sqrt 2} (i +j)$

But.. answer is option (a). Please explain why modulus is not taken. Or I am missing some concept in vectors.

Answer 1

Your mistake seems to be in writing $|x|$ as being the same as $\pm x$ which is not true.

So answer a) is correct