Question

Find a unit vector perpendicular to the vectors `hat"j" + 2hat"k"`  and  `hat"i" + hat"j"`.

Answer 1

Let `bar"a" = hat"j" + 2hat"k"` and `bar"b" = hat"i" + hat"j"`

Then `bar"a" xx bar"b" = |(hat"i",hat"j",hat"k"),(0,1,2),(1,1,0)|`

= `(0 - 2)hat"i" - (0 - 2)hat"j" + (0 - 1)hat"k"`

= `- 2hat"i" + 2hat"j" - hat"k"`

∴ `|bar"a"xxbar"b"| = sqrt((-2)^2 + 2^2 + (- 1)^2)`

= `sqrt(4 + 4 + 1)`

= `sqrt(9)`

= 3

∴ Unit vector perpendicular to both `bar"a"` and `bar"b"`

= `+- (bar"a" xx bar"b")/(|bar"a" xx bar"b"|) = +- ((- 2hat"i" + 2hat"j" - hat"k")/3)`

= `+- (2/3hat"i" - 2/3hat"j" + 1/3hat"k")`