Question

Find a vector of magnitude 4 units perpendicular to each of the vectors `2hat i - hat j + hat k and hat i + hat j - hat k`  and hence verify your answer.

Answer 1

Given vectors are: 

`overset->a = 2 hat i - hat j + hat k`

`overset->b = hat i - hat j + hat k` 

`overset->a xx overset->b = |(hat i, hat j, hat k),(2, -1, 1),(1, 1, -1)|`

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

= `3 hat j + 3 hat k`

Now, `|overset->a xx overset->b|`

= `sqrt((3)^2 + (3)^2)`

= `sqrt(9 + 9)`

= `sqrt18 = 3sqrt2`

Therefore required vector is, `4((overset->a xx overset-> b)/(|overset->a xx overset->b|))`

= `4/(3sqrt2)(3hat j + 3 hat k)`

= `(2sqrt2)/3 xx 3(hat j + hat k)`

= `2sqrt2 (hat j + hat k)`