If points A,B and C with position vectors `2 hat(i) - hat(j) + hat(k) , hat(i) - 3 hat(j) - 5hat(k)` and `alpha hat(i) - 3 hat(j) + hat(k)` respectively are the vertices of a right-anged triangle with `/_C = ( pi )/( 2)`, then the values of `alpha ` are

Options

A. 2,-1
B. 2,1
C. `-2,1`
D. `-2,-1`

Answer

To solve the problem, we need to find the values of \( \alpha \) such that the points A, B, and C form a right-angled triangle with the right angle at point C. The position vectors of the points A, B, and C are given as follows:

- \( \vec{A} = 2\hat{i} - \hat{j} + \hat{k} \)
- \( \vec{B} = \hat{i} - 3\hat{j} - 5\hat{k} \)
- \( \vec{C} = \alpha\hat{i} - 3\hat{j} + \hat{k} \)

### Step 1: Find the vectors \( \vec{CA} \) and \( \vec{CB} \)

The vector \( \vec{CA} \) can be calculated as:
\[ \vec{CA} = \vec{A} - \vec{C} = (2\hat{i} - \hat{j} + \hat{k}) - (\alpha\hat{i} - 3\hat{j} + \hat{k}) = (2 - \alpha)\hat{i} + (2)\hat{j} + (0)\hat{k} \]

The vector \( \vec{CB} \) can be calculated as:
\[ \vec{CB} = \vec{B} - \vec{C} = (\hat{i} - 3\hat{j} - 5\hat{k}) - (\alpha\hat{i} - 3\hat{j} + \hat{k}) = (1 - \alpha)\hat{i} + (0)\hat{j} - (6)\hat{k} \]

### Step 2: Use the condition for right angles

For the triangle to be right-angled at C, the dot product of vectors \( \vec{CA} \) and \( \vec{CB} \) must be zero:
\[ \vec{CA} \cdot \vec{CB} = 0 \]

Calculating the dot product:
\[ \vec{CA} \cdot \vec{CB} = ((2 - \alpha)\hat{i} + 2\hat{j} + 0\hat{k}) \cdot ((1 - \alpha)\hat{i} + 0\hat{j} - 6\hat{k}) \]
\[ = (2 - \alpha)(1 - \alpha) + 2(0) + 0(-6) \]
\[ = (2 - \alpha)(1 - \alpha) \]

### Step 3: Set the dot product to zero

Setting the dot product equal to zero:
\[ (2 - \alpha)(1 - \alpha) = 0 \]

### Step 4: Solve for \( \alpha \)

This equation gives us two cases:
1. \( 2 - \alpha = 0 \) → \( \alpha = 2 \)
2. \( 1 - \alpha = 0 \) → \( \alpha = 1 \)

### Conclusion

Thus, the values of \( \alpha \) are \( 2 \) and \( 1 \).