Answer
$\frac{\sqrt 3} { 3}i-\frac{\sqrt 3} {3}j-\frac{\sqrt 3} {3}k$ and
$-\frac{\sqrt 3} { 3}i+\frac{\sqrt 3} {3}j+\frac{\sqrt 3} {3}k$
Work Step by Step
$j-k = \lt 0,1,-1 \gt$ and $i+j = \lt 1,1,0 \gt$
Cross product will produce a vector orthogonal to both.
$v=\lt 0,1,-1 \gt \times \lt 1,1,0 \gt = \lt 1,-1,-1 \gt$
$|v|=\sqrt {1^2+(-1)^2+(-1)^2}=\sqrt 3$
$\frac {v}{|v|}=\lt \frac{1} {\sqrt 3},-\frac{1} {\sqrt 3},-\frac{1} {\sqrt 3}\gt$
$=\frac{\sqrt 3} { 3}i-\frac{\sqrt 3} {3}j-\frac{\sqrt 3} {3}k$
The other orthogonal vector is: $=-\frac{\sqrt 3} { 3}i+\frac{\sqrt 3} {3}j+\frac{\sqrt 3} {3}k$
