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

Options

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

Answer

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 will follow these steps:

### Step 1: Understand the condition for orthogonality
Two vectors are orthogonal if their dot product is zero. Thus, we need to calculate the dot product of the given vectors and set it equal to zero.

### Step 2: Write down the vectors
Let:
- Vector A = \(\hat{i} - x \hat{j} - 2 \hat{k}\)
- Vector B = \(2 \hat{i} + \hat{j} + y \hat{k}\)

### Step 3: Calculate the dot product
The dot product of two vectors \(A\) and \(B\) is given by:
\[ A \cdot B = (1)(2) + (-x)(1) + (-2)(y) \]
Calculating this gives:
\[ A \cdot B = 2 - x - 2y \]

### Step 4: Set the dot product equal to zero
Since the vectors are orthogonal, we set the dot product equal to zero:
\[ 2 - x - 2y = 0 \]

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

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

### Final Result
The locus of the point (x, y) for which the given vectors are orthogonal is a straight line given by the equation:
\[ x + 2y = 2 \]

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