The values of `a` for which the points `A, B,` and `C` with position vectors `2hati - hatj + hatk, hati - 3hatj - 5hatk,` and `ahati - 3hatj + hatk`,respectively, are the vertices of a right-angled triangle with `C = pi/2` are

Options

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

Answer

To solve the problem, we need to find the values of \( a \) for which the points \( A, B, \) and \( C \) with position vectors \( \mathbf{A} = 2\hat{i} - \hat{j} + \hat{k} \), \( \mathbf{B} = \hat{i} - 3\hat{j} - 5\hat{k} \), and \( \mathbf{C} = a\hat{i} - 3\hat{j} + \hat{k} \) form a right-angled triangle at point \( C \).

### Step 1: Find the vectors \( \mathbf{AC} \) and \( \mathbf{BC} \)

1. **Calculate vector \( \mathbf{AC} \)**:
\[ \mathbf{AC} = \mathbf{C} - \mathbf{A} = (a\hat{i} - 3\hat{j} + \hat{k}) - (2\hat{i} - \hat{j} + \hat{k}) \]
Simplifying this:
\[ \mathbf{AC} = (a - 2)\hat{i} + (-3 + 1)\hat{j} + (1 - 1)\hat{k} = (a - 2)\hat{i} - 2\hat{j} + 0\hat{k} \]
Thus,
\[ \mathbf{AC} = (a - 2)\hat{i} - 2\hat{j} \]

2. **Calculate vector \( \mathbf{BC} \)**:
\[ \mathbf{BC} = \mathbf{C} - \mathbf{B} = (a\hat{i} - 3\hat{j} + \hat{k}) - (\hat{i} - 3\hat{j} - 5\hat{k}) \]
Simplifying this:
\[ \mathbf{BC} = (a - 1)\hat{i} + (-3 + 3)\hat{j} + (1 + 5)\hat{k} = (a - 1)\hat{i} + 0\hat{j} + 6\hat{k} \]
Thus,
\[ \mathbf{BC} = (a - 1)\hat{i} + 6\hat{k} \]

### Step 2: Use the dot product to find the condition for a right angle

Since \( \angle C \) is \( 90^\circ \), the vectors \( \mathbf{AC} \) and \( \mathbf{BC} \) must be perpendicular. Therefore, their dot product must equal zero:
\[ \mathbf{AC} \cdot \mathbf{BC} = 0 \]

Calculating the dot product:
\[ \mathbf{AC} \cdot \mathbf{BC} = ((a - 2)\hat{i} - 2\hat{j}) \cdot ((a - 1)\hat{i} + 6\hat{k}) \]
Using the properties of the dot product:
\[ = (a - 2)(a - 1) + (-2)(0) + (0)(6) = (a - 2)(a - 1) \]

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

### Step 3: Solve the equation

The solutions to the equation are:
1. \( a - 2 = 0 \) → \( a = 2 \)
2. \( a - 1 = 0 \) → \( a = 1 \)

Thus, the values of \( a \) for which the points \( A, B, \) and \( C \) form a right-angled triangle at \( C \) are:
\[ \boxed{1 \text{ and } 2} \]