Dot product of a vector with ` hat i+ hat j-3 hat k , hat i+3 hat j-2 hat k` and `2 hat i+ hat j+4 hat k` are `0,5` and `8` respectively. Find the vector. Dot products of a vector with vectors ` hat i- hat j+ hat k ,2 hat i+ hat j-3 hat k` and ` hat i+ hat j+ hat k` are respectively `4,0` and `2.` Find the vector.

Answer

`(hat(i)+2hat(j)+hat(k))`

Explanation

Let the given vectors be `vec(a), vec(b)` and `vec(c)` respectively and let the required vector be `vec(d)=d_(1) vec(i)+d_(2) hat(j)+d_(3) hat(k)`.
Then, `vec(d).vec(a)=0, vec(d).vec(b)=5` and `vec(d).vec(c)=8`
`implies d_(1)+d_(2)-3d_(3)=0` ...(i), `d_(1)+3d_(2)-2d_(3)=5` ...(ii) and `2d_(1)+d_(2)+4d_(3)=8` ...(iii)
On subtracting (i) from (i) from (ii), we get `2d_(2)+d_(3)=5`. ...(iv)
On multiplying (i) by 2 and substraction (iii) from it, we get `d_(2)-10 d_(3)=-8`. ...(v)
On solving (iv) and (v), we get `d_(2)=2, d_(3)=1` and therefore, `d_(1)=1`.