Question

Find a vector which is perpendicular to both vector A and B, where A=2i+3j+4k B=i+2j+3k?

Answer 1

The vector is `=〈1,-2,1〉`

Explanation:

The vector perpendicular to 2 vectors is calculated with the determinant (cross product)

`| (veci,vecj,veck), (d,e,f), (g,h,i) |`

where `〈d,e,f〉` and `〈g,h,i〉` are the 2 vectors

Here, we have `veca=〈2,3,4〉` and `vecb=〈1,2,3〉`

Therefore,

`| (veci,vecj,veck), (2,3,4), (1,2,3) |`

`=veci| (3,4), (2,3) | -vecj| (2,4), (1,3) | +veck| (2,3), (1,2) |`

`=veci(3*3-2*4)-vecj(2*3-1*4)+veck(2*2-3*1)`

`=〈1,-2,1〉=vecc`

Verification by doing 2 dot products

`〈1,-2,1〉.〈2,3,4〉=1*2-2*+1*4=0`

`〈1,-2,1〉.〈1,2,3〉=1*1-2*2+1*3=0`

So,

`vecc` is perpendicular to `veca` and `vecb`

Answer 2

`i-2j+k`

Explanation:

Cross product of vectors A and B is perpendicular to each vector A and B.

`vecA`x `vecB=i(9-8)-j(6-4)+k(4-3)`

`vecA`x `vecB=i-2j+k`

Answer 3

We know that cross product of any two vectors yields a vector which is perpendicular to both vectors
`:.` for two vectors `vecA and vecB` if `vecC` is the vector perpendicular to both.
`vecC=vecAxxvecB=` `[(hati, hatj, hatk), (A_1, A_2,A_3),(B_1, B_2, B_3)]`
`=(A_2B_3−B_2A_3)hati−(A_1B_3−B_1A_3)hatj+(A_1B_2−B_1A_2)hatk`.
Inserting given vectors we obtain
`vecC=` `[(hati, hatj, hatk), (2, 3,4),(1, 2, 3)]`
`=(3xx3-2xx4)hati−(2xx3−1xx4)hatj+(2xx2−1xx3)hatk`.
`=hati−2hatj+hatk`.