The unit vector orthogonal to vector ` -hat i+ hat j+2 hat k` and making equal angles with the x and y-axis a.`+-1/3(2 hat i+2 hat j- hat k)` b. `+-1/3( hat i+ hat j- hat k)` c. `+-1/3(2 hat i-2 hat j- hat k)` d. none of these

Options

A. `+-1/3(2hati+2hatj-hatk)`
B. `19/(5sqrt43)`
C. `+-1/3(hati+hatj-hatk)`
D. none of these

Answer

a

Explanation

Let l, m and n be the direaction consines of the required vector. Then, l-m (given ) therefore,
Required vector `vecr = l hati + mhatj + nhatk = lhati + lhatj + n hatk`
now, `l^(2) + m^(2) +n^(2)= 1 or 2l^(2) + n^(2) =1`
since `vecr` is perpendicular to `- hati + 2hatj + 2hatk`,
`vecr . ( -hati + 2hatj + 2hatk) =0`
`Rightarrow -l + 2l + 2n =0 or l + 2n =0`
Form (i) and (ii) , we get `n = +- 1/3 , l = +- 2/3`
Hence, required vector
`vecr = 1/3 (+- 2hati +- 2hatj -+ hatk) = +- 1/3 (2 hati + 2hatj -hatk) `