A unit vector which is perpendicular to the vector 2 i- j+2 k and is coplanar with the vec…

Question

A unit vector which is perpendicular to the vector $2\hat i-\hat j+2\hat k$ and is coplanar with the vectors $\hat i+\hat j-\hat k$ and $2\hat i+2\hat j-\hat k$ is:

Accepted Answer

Let the vector be $p(\hat i+\hat j-\hat k)+q(2\hat i+2\hat j-\hat k)=(p+2q)\hat i+(p+2q)\hat j-(p+q)\hat k$. Perpendicularity to $2\hat i-\hat j+2\hat k$ gives:

$$2(p+2q)-(p+2q)-2(p+q)=0 \implies -p=0 \implies p=0$$

So the vector is $2q\hat i+2q\hat j-q\hat k$. Making it a unit vector: $4q^2+4q^2+q^2=1 \implies q=\pm\dfrac13$.

Taking $q=\dfrac13$: $\dfrac23\hat i+\dfrac23\hat j-\dfrac13\hat k = \dfrac{2\hat i+2\hat j-\hat k}3$

Solutions · 1