Question

If `bar"a" = 2hat"i" + hat"j" - 3hat"k"` and  `bar"b" = hat"i" - 2hat"j" + hat"k"`, find a vector of magnitude 5 perpendicular to both `bar"a"` and `bar"b"`.

Answer 1

Given: `bar"a" = 2hat"i" + hat"j" - 3hat"k"` and 

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

∴ `bar"a" xx bar"b" = |(hat"i", hat"j", hat"k"),(2,1,-3),(1,-2,1)|`

`= (1 - 6)hat"i" - (2 + 3)hat"j" + (- 4 - 1)hat"k"`

`= - 5hat"i" - 5hat"j" - 5hat"k"`

∴ `|bar"a" xx bar"b"| = sqrt((-5)^2 + (- 5)^2 + (- 5)^2)`

`= sqrt(25 + 25 +25) = sqrt75 = 5sqrt3`

∴ unit vectors perpendicular to both the vectors `bar"a"` and `bar"b"`

`hatn = +-((bar"a"xxbar"b"))/(|bar"a" xx bar"b"|)`

`hatn = +- ((- 5hat"i" - 5hat"j" - 5hat"k"))/(5sqrt3)`

`hatn = +- ((-5)(hati + hatj + hatk))/(5sqrt3)` 

`hatn = -+ ((hat"i" + hat"j" + hat"k"))/sqrt3`

∴ required vectors of magnitude 5 units

`= +- 5/sqrt3 (hat"i" + hat"j" + hat"k")`