O is the origin and P is point at a distance of 3 units from origin. If direction ratios of OP are `lt 1,-2,-2 gt`, then the coordinates of P are

Options

A. `(1,-2,-2)`
B. `(3,-6,-6)`
C. `((1)/(3), - (2)/(3), -(2)/(3) )`
D. `((1)/(9) ,-(2)/(9),-(2)/(9))`

Answer

To find the coordinates of point P, we can follow these steps:

### Step 1: Understanding the given information
We know that:
- O is the origin (0, 0, 0).
- P is a point at a distance of 3 units from the origin.
- The direction ratios of OP are given as (1, -2, -2).

### Step 2: Finding the magnitude of the direction ratios
The direction ratios can be represented as a vector OP:
\[ \text{OP} = \langle 1, -2, -2 \rangle \]
Now, we can find the magnitude of this vector:
\[ \text{Magnitude of OP} = \sqrt{1^2 + (-2)^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Step 3: Finding the unit vector in the direction of OP
To find the unit vector in the direction of OP, we divide each component of the direction ratios by the magnitude:
\[ \text{Unit vector} = \frac{1}{3} \langle 1, -2, -2 \rangle = \left\langle \frac{1}{3}, -\frac{2}{3}, -\frac{2}{3} \right\rangle \]

### Step 4: Finding the coordinates of point P
Since the distance OP is given as 3 units, we can find the coordinates of P by multiplying the unit vector by the distance:
\[ \text{Coordinates of P} = 3 \times \left\langle \frac{1}{3}, -\frac{2}{3}, -\frac{2}{3} \right\rangle = \left\langle 1, -2, -2 \right\rangle \]

### Conclusion
Thus, the coordinates of point P are:
\[ P(1, -2, -2) \]