The vectors `vec(a)=xvec(i)+y vec(j)+zvec(k), vec(b)=hat(k), vec(c)` are such that they form a right handed system. What is `vec(c)` equal to ?

Options

A. `hat(j)`
B. `y hat(j)-x hat(k)`
C. `y hat(i)-x hat(j)`
D. `x hat(i)-y hat(j)`

Answer

C

Explanation

We know, scalar triple product `(vec(a)xxvec(b)).vec(c)` is positive or negative according as `vec(a), vec(b), vec(c)` from a right handed or left handed system respectively.
consider option (a)
Let `vec(c)=vec(j)`
`:. [vec(a)vec(b)vec(c)]=|(x,y,z),(0, 0,1),(0,1,0)|=x(-1)-y(0)+z(0)`
`=-x`
option (b)
Let `vec(c)=yhat(j)-xhat(k)`
`:. [vec(a)vec(b)vec(c)]=|(x,y,z),(0,0,1),(0,y,-x)|=x(-y)-y(0)+z(0)`
`=-xy`
option (c)
Let `vec(c)=yhat(i)-xhat(j)`
`:. [vec(a)vec(b)vec(c)]=|(x,y,z),(0,0,1),(y,-x,0)|`
`=x(x)-y(-y)+z(0)=x^(2)+y^(2)`
Since, scalar triple product is positive when
`vec(c)=y hat(i)-x hat(j)`
`:.` option (c) is correct.