Question

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

Answer 1

`+-(3hati-3hatj+hatk)/(sqrt19`

Explanation:

If `vecA=hati+hatj and vecB =2hati+hatj-3hatk`
then vectors which will be normal to the plane containing `vec A and vecB` are either`vecAxxvecB or vecBxxvecA` .So we are to find out the unit vectors of these two vector . One is opposite to another.

Now `vecAxxvecB=(hati+hatj+0hatk )xx(2hati+hatj-3hatk)`
`=(1*(-3)-0*1)hati+(0*2-(-3)*1)hatj+(1*1-1*2)hatk`
`=-3hati+3hatj-hatk`

So unit vector of `vecAxxvecB=(vecAxxvecB)/|vecAxxvecB|`
`=-(3hati-3hatj+hatk)/(sqrt(3^2+3^2+1^2))=-(3hati-3hatj+hatk)/(sqrt19`

And unit vector of `vecBxxvecA=+(3hati-3hatj+hatk)/sqrt19`