What is the locus of the point (x, y) for which the vectors `(hat(i)-xhat(j)-2hat(k)) and (2hat(i)+hat(j)+yhat(k))` are orthogonal?

Options

A. A circle
B. An ellipse
C. A parabola
D. A straight line

Explanation

To find the locus of the point (x, y) for which the vectors \( \hat{i} - x\hat{j} - 2\hat{k} \) and \( 2\hat{i} + \hat{j} + y\hat{k} \) are orthogonal, we follow these steps:

### Step 1: Set up the vectors
We have two vectors:
1. \( \mathbf{A} = \hat{i} - x\hat{j} - 2\hat{k} \)
2. \( \mathbf{B} = 2\hat{i} + \hat{j} + y\hat{k} \)

### Step 2: Use the condition for orthogonality
Two vectors are orthogonal if their dot product is zero. Therefore, we need to compute the dot product \( \mathbf{A} \cdot \mathbf{B} \) and set it equal to zero.

### Step 3: Calculate the dot product
The dot product \( \mathbf{A} \cdot \mathbf{B} \) is calculated as follows:
\[ \mathbf{A} \cdot \mathbf{B} = (1)(2) + (-x)(1) + (-2)(y) \]
This simplifies to:
\[ 2 - x - 2y \]

### Step 4: Set the dot product to zero
To find the locus, we set the dot product equal to zero:
\[ 2 - x - 2y = 0 \]

### Step 5: Rearrange the equation
Rearranging the equation gives:
\[ x + 2y = 2 \]

### Step 6: Identify the locus
The equation \( x + 2y = 2 \) represents a straight line in the xy-plane.

### Final Answer
The locus of the point (x, y) is the straight line given by the equation:
\[ x + 2y - 2 = 0 \]