The co-ordinates of the point P on the line
`vecr =(hati +hatj + hatk)+lambda (-hati +hatj -hatk)` which is nearest to the
origin is

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

Explanation

To find the coordinates of the point P on the line given by the equation \(\vec{r} = \hat{i} + \hat{j} + \hat{k} + \lambda (-\hat{i} + \hat{j} - \hat{k})\) that is nearest to the origin, we will follow these steps:

### Step 1: Write the equation of the line
The line can be expressed as:
\[ \vec{r} = (1 - \lambda)\hat{i} + (1 + \lambda)\hat{j} + (1 - \lambda)\hat{k} \]
This means that the coordinates of any point P on the line can be represented as:
\[ P(\lambda) = (1 - \lambda, 1 + \lambda, 1 - \lambda) \]

### Step 2: Define the position vector of the origin
The position vector of the origin O is:
\[ \vec{O} = 0\hat{i} + 0\hat{j} + 0\hat{k} = \vec{0} \]

### Step 3: Define the vector OP
The vector from the origin O to the point P on the line is given by:
\[ \vec{OP} = P - O = (1 - \lambda)\hat{i} + (1 + \lambda)\hat{j} + (1 - \lambda)\hat{k} \]

### Step 4: Find the direction vector of the line
The direction vector of the line can be extracted from the equation:
\[ \vec{d} = -\hat{i} + \hat{j} - \hat{k} \]

### Step 5: Set up the condition for perpendicularity
For the point P to be the nearest point to the origin, the vector \(\vec{OP}\) must be perpendicular to the direction vector \(\vec{d}\). This gives us the dot product condition:
\[ \vec{OP} \cdot \vec{d} = 0 \]

### Step 6: Compute the dot product
Substituting \(\vec{OP}\) and \(\vec{d}\):
\[ ((1 - \lambda)\hat{i} + (1 + \lambda)\hat{j} + (1 - \lambda)\hat{k}) \cdot (-\hat{i} + \hat{j} - \hat{k}) = 0 \]
Calculating the dot product:
\[ -(1 - \lambda) + (1 + \lambda) - (1 - \lambda) = 0 \]
This simplifies to:
\[ -1 + \lambda + 1 + \lambda - 1 + \lambda = 0 \]
Combining like terms:
\[ 3\lambda - 1 = 0 \]

### Step 7: Solve for \(\lambda\)
From the equation \(3\lambda - 1 = 0\), we solve for \(\lambda\):
\[ 3\lambda = 1 \implies \lambda = \frac{1}{3} \]

### Step 8: Substitute \(\lambda\) back to find point P
Now, substitute \(\lambda = \frac{1}{3}\) back into the equation for P:
\[ P\left(\frac{1}{3}\right) = \left(1 - \frac{1}{3}, 1 + \frac{1}{3}, 1 - \frac{1}{3}\right) = \left(\frac{2}{3}, \frac{4}{3}, \frac{2}{3}\right) \]

### Final Answer
The coordinates of the point P on the line that is nearest to the origin are:
\[ \left(\frac{2}{3}, \frac{4}{3}, \frac{2}{3}\right) \]