The coordinates of the point `P` on the line ` vec r=( hat i+ hat j+ hat k)+lambda(- hat i+ hat j- hat k)` which is nearest to the origin is a. `(2/4,4/3,2/3)` b. `(-2/3,-4/3,2/3)` c. `(2/3,-4/3,2/3)` d. none of these

Options

A. `((2)/(3)(4)/(3),(2)/(3))`
B. `(-(2)/(3)-(4)/(3),(2)/(3))`
C. `((2)/(3)(4)/(3),-(2)/(3))`
D. none of these

Answer

a

Explanation

Let the point `P` be `(x, y, z)`, then the vector `xhati+yhatj+zhatk` will lie on the line. Thus,
`" "(x-1)hati+(y-1)hatj+ (z-1)hatk= -lamdahati+lamdahatj-lamdahatk`
`rArr" "x=1-lamda, y=1+lamda and z=1-lamda`
Now point P is nearest to the origin
`rArr" "D= (1-lamda)^(2) + (1+lamda)^(2) + (1-lamda)^(2)`
or `" "(dD)/(dlamda)= -4(1-lamda)+ 2(1+lamda) = 0 rArr lamda = (1)/(3)`
Hence, the point is `((2)/(3), (4)/(3), (2)/(3))`