Let `alpha, beta and gamma` be distinct real numbers. The points with position vectors `alpha hat(i) + beta hat(j) + gamma hat(k), beta hat(i) + gamma hat(j) + alpha hat(k) and gamma hat(i) + alpha hat(j) + beta hat(k)`

Options

A. A. Are collinear
B. B. Form an equilateral triangle
C. C. Form a scalene triangle
D. D. Form a right-angled triangle

Answer

To determine the relationship between the points with the given position vectors, we will follow these steps:

### Step 1: Define the Position Vectors
Let:
- \( A = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \)
- \( B = \beta \hat{i} + \gamma \hat{j} + \alpha \hat{k} \)
- \( C = \gamma \hat{i} + \alpha \hat{j} + \beta \hat{k} \)

### Step 2: Find the Vector \( \overrightarrow{AB} \)
The vector \( \overrightarrow{AB} \) can be computed as:
\[ \overrightarrow{AB} = B - A = (\beta - \alpha) \hat{i} + (\gamma - \beta) \hat{j} + (\alpha - \gamma) \hat{k} \]

### Step 3: Calculate the Magnitude of \( \overrightarrow{AB} \)
The magnitude of \( \overrightarrow{AB} \) is given by:
\[ |\overrightarrow{AB}| = \sqrt{(\beta - \alpha)^2 + (\gamma - \beta)^2 + (\alpha - \gamma)^2} \]

### Step 4: Find the Vector \( \overrightarrow{BC} \)
Now, compute the vector \( \overrightarrow{BC} \):
\[ \overrightarrow{BC} = C - B = (\gamma - \beta) \hat{i} + (\alpha - \gamma) \hat{j} + (\beta - \alpha) \hat{k} \]

### Step 5: Calculate the Magnitude of \( \overrightarrow{BC} \)
The magnitude of \( \overrightarrow{BC} \) is:
\[ |\overrightarrow{BC}| = \sqrt{(\gamma - \beta)^2 + (\alpha - \gamma)^2 + (\beta - \alpha)^2} \]

### Step 6: Find the Vector \( \overrightarrow{CA} \)
Next, compute the vector \( \overrightarrow{CA} \):
\[ \overrightarrow{CA} = A - C = (\alpha - \gamma) \hat{i} + (\beta - \alpha) \hat{j} + (\gamma - \beta) \hat{k} \]

### Step 7: Calculate the Magnitude of \( \overrightarrow{CA} \)
The magnitude of \( \overrightarrow{CA} \) is:
\[ |\overrightarrow{CA}| = \sqrt{(\alpha - \gamma)^2 + (\beta - \alpha)^2 + (\gamma - \beta)^2} \]

### Step 8: Compare the Magnitudes
Now we compare the magnitudes:
- \( |\overrightarrow{AB}| = |\overrightarrow{BC}| = |\overrightarrow{CA}| \)

Since all three sides are equal, we conclude that the points \( A, B, C \) form an equilateral triangle.

### Final Answer
Thus, the points with the given position vectors form an equilateral triangle.

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