1.

A vector of magnitude 4 which is equally inclined to the vectors `hati +hatj,hatj +hatk and hatk +hati`, isA. `4/sqrt3 (hati -hatj -hatk)`B. ` 4/sqrt3 (hati +hatj -hatk)`C. ` 4/sqrt3 (hati +hatj +hatk)`D. `4/sqrt3 (hati +hatj -hatk)`

Answer» Correct Answer - C
Let the required vector be ` vecr = xhati + yhaty + zhatk` , Then
` |vecr| =4 Rightarrow x^(2) +y^(2) +z^(2) =16`
Now, `vecr`, sequally inclined to the vectors `hati +hatj ,hatj + hatk and hati`
` (vecr. (hati + hatj)/(|vecr|sqrt2)= (vecr.(hatj +hatk))/(|vecr|sqrt2) = (vecr.(hatj+hati))/(|vecr|sqrt2)`
` Rightarrow x + y=y +z =z +x =lambda (say) `
` Rightarrow 2(x +y +z) =3 lambda Rightarrow x+y +z = (3 lambda)/2`
Now, ` x + y+ lambda and x +y + z = (3lambda)/2 Rightarrow z = lamnbda/2`
Similarly, we have, ` x=y = lambda/2`
Substituting these values in (i) , we get ` lambda = +- 8/sqrt3`
Hence, ` vecr= -+ 8/(2sqrt3) ( hati +hatj +hatk) = -+ 4/sqrt3 (hati +hatj + hatk)`


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