The equation `vecr = lamda hati + mu hatj` represents :

Options

A. The Z-axis
B. the plane
C. the X-axis
D. the Y-axis

Answer

To solve the problem, we need to analyze the given equation:

\[ \vec{r} = \lambda \hat{i} + \mu \hat{j} \]

### Step 1: Understand the components of the equation
In this equation:
- \(\vec{r}\) represents the position vector of a point in three-dimensional space.
- \(\lambda\) and \(\mu\) are parameters that can take any real values.
- \(\hat{i}\) and \(\hat{j}\) are the unit vectors in the x-direction and y-direction, respectively.

### Step 2: Interpret the equation
The equation indicates that the position vector \(\vec{r}\) can be expressed as a linear combination of the unit vectors \(\hat{i}\) and \(\hat{j}\). This means that any point represented by \(\vec{r}\) lies in the x-y plane, where:
- \(\lambda\) determines the x-coordinate,
- \(\mu\) determines the y-coordinate.

### Step 3: Determine the implications
Since there are no terms involving \(\hat{k}\) (the unit vector in the z-direction), the z-coordinate is not defined in this equation. This implies that the z-coordinate can take any value, meaning that the points described by this equation lie in a plane parallel to the x-y plane.

### Step 4: Conclusion
Thus, the equation \(\vec{r} = \lambda \hat{i} + \mu \hat{j}\) represents a plane in three-dimensional space, specifically the x-y plane, where the z-coordinate can be any value.

### Final Answer:
The equation represents a plane.

---