Express `-hat(i)-3hat(j)+4hat(k)` as the linear combination of the vectors `2hat(i)+hat(j)-4hat(k)`, `2hat(i)-hat(j)+3hat(k)` and `3hat(i)+hat(j)-2hat(k)`.

Explanation

Linear combination of vectors are
`-hat("i")-3hat(j)+4hat(k)=x(2hat("i")+hat(j)-4hat(k))+y(2hat("i")-hat(j)+3hat(k))+z(3hat("i")+hat(j)-2hat(k))` ...(i)
Here x,y,z are constant.
`-hat("i")-3hat(j)+4hat(k)=2xhat("i")+xhat(j)-4xhat(k)+2yhat("i")-yhat(j)+3yhat(k)+3zhat("i")+zhat(j)-2zhat(k)`
`=hat("i")(2x+2y+3z)+hat(j)(x-y+z)+hat(k)(-4x+3y-2z)`
On comparing the coefficient of `hat("i"),hat(j)andhat(k)`, we get
`-1=2x+2y+3z` ...(ii)
`-3=x-y+z` ...(iii)
`4=-4x+3y-2z` ...(iv)
By equations (ii) and (iii),
`(2x+2y+3z=-1)xx1`
`(x-y+z=-3)xx2`
Then `{:(2x+2y+3z=-1),(2x-2y+2z=-6),(ul(-" "+" " -" "+)),(" "4y+z=5):}` ...(v)
By equations (ii) and (iv),
`{:((2x+2y+3z=-1)xx2),((-4x+3y-2z=4)xx1),(4x+4y+6z=-2),(ul(-4x+3y-2z=4)),(" "7y+4z=2):}` ...(vi)
By equations (v) and (vi),
`{:((4y+z=5)xx4),((7y+4z=2)xx1),(16y+4z=20),(" "7y+4z=2),(ul(-" "-" "-)),(-9y=18),(" "y=-2):}`
Put the value of y in equation (v) ,we get
`8+z=5`
`z=-3`
Put the value of y and z in equation (iii) ,we get
x-2-3=-3
x=-3+5=2
Put the value of x,y,z in equation (i) ,we get
`-hat("i")-3hat(j)+4hat(k)=2(2hat("i")+hat(j)-4hat(k))-2(2hat("i")-hat(j)+3hat(k))`
`-3(3hat("i")+hat(j)-2hat(k))`