Question

What is the unit vector that is orthogonal to the plane containing `( i - 2 j + 3 k)` and `( - 4 i - 5 j + 2 k)`?

Answer 1

The unit vector is `((11veci)/sqrt486-(14vecj)/sqrt486-(13veck)/sqrt486)`

Explanation:

Firstly, we need the vector perpendicular to other two vectros:
For this we do the cross product of the vectors:
Let `vecu=〈1,-2,3〉` and `vecv=〈-4,-5,2〉`
The cross product `vecu`x`vecv` `=`the determinant
`∣((veci,vecj,veck),(1,-2,3),(-4,-5,2))∣`
`=veci∣((-2,3),(-5,2))∣-vecj∣((1,3),(-4,2))∣+veck∣((1,-2),(-5,-5))∣`
`=11veci-14vecj-13veck`
So `vecw=〈11,-14,-13〉`
We can check that they are perpendicular by doing the dot prodct.
`vecu.vecw=11+28-39=0`
`vecv.vecw=-44+70-26=0`
The unit vector `hatw=vecw/(∥vecw∥)`
The modulus of `vecw=sqrt(121+196+169)=sqrt486`
So the unit vector is `((11veci)/sqrt486-(14vecj)/sqrt486-(13veck)/sqrt486)`