Find the cartesian form of the equation of the plane .
`bar(r)=(hat(i)+hat(j))+s(hat(i)-hat(j)+2hat(k))+t(hat(i)+2hat(j)+hat(k))`

Explanation

The vector equation of the plane.
`bar(r)=bar(a)+ s bar(b)+t bar(c)` is `bar(r)*(bar(b)xx bar(c))=bar(a)*(bar(b)xx bar(c))`
Here, `bar(a)=hat(i)+hat(j),bar(b)=hat(i)-hat(j)+2hat(k), c=hat(i)+2hat(j)+hat(k)`
`:.bar(b)xxbar(c)=|(hat(i),hat(j),hat(k)),(1,-1,2),(1,2,1)|`
`=(-1-4)hat(i)-(1-2)hat(j)+(2+1)hat(k)`
`=-5hat(i)+hat(j)+3hat(k)`
`:.bar(a)*(bar(a)xxbar(c))=(hat(i)+hat(j))*(-5hat(i)+hat(j)+3hat(k))`
`=(-5)+1+0=-4`
`:.` The vector equation of the given plane is
`:.bar(r)*(-5hat(i)+hat(j)+3hat(k))=-4`
If `bar(r)=x hat(i)+y hat(j) + z hat(k)` then the above equation becomes
`(xhat(i)+y hat(j)+z hat(k))*(-5hat(i)+hat(j)+3hat(k))=-4`
`:.x(-5)+y(1)+z(3)=-4`
`:.-5x+y+3z=-4`
This is the cartesian form of the given equation of the given plane.