The values of 'a', for which the points A, B, C with position vectors $2\hat{i}-\hat{j}+\hat{k}$, $\hat{i}-3\hat{j}-5\hat{k}$ and $a\hat{i}-3\hat{j}+\hat{k}$ respectively are the vertices of right-angled triangle with $\angle C=\frac{\pi}{2}$ are:

Question

Accepted Answer

To determine the values of 'a' for which the points A, B, and C form a right-angled triangle with the right angle at C, we need to use the concept of the dot product of vectors. The position vectors of the points are given as follows:

- A: $ \vec{A} = 2\hat{i} - \hat{j} + \hat{k} $
- B: $ \vec{B} = \hat{i} - 3\hat{j} - 5\hat{k} $
- C: $ \vec{C} = a\hat{i} - 3\hat{j} + \hat{k} $

For $\angle C$ to be a right angle, the vectors $\vec{CA}$ and $\vec{CB}$ must be perpendicular. This means their dot product should be zero.

First, calculate the vectors $\vec{CA}$ and $\vec{CB}$:

$$
\vec{CA} = \vec{A} - \vec{C} = (2 - a)\hat{i} + 2\hat{j} + 0\hat{k}
$$

$$
\vec{CB} = \vec{B} - \vec{C} = (1 - a)\hat{i} + 0\hat{j} - 6\hat{k}
$$

The dot product $\vec{CA} \cdot \vec{CB}$ is given by:

$$
\vec{CA} \cdot \vec{CB} = [(2 - a)\hat{i} + 2\hat{j} + 0\hat{k}] \cdot [(1 - a)\hat{i} + 0\hat{j} - 6\hat{k}]
$$

$$
= (2 - a)(1 - a) + 2 \times 0 + 0 \times (-6)
$$

$$
= (2 - a)(1 - a)
$$

For the vectors to be perpendicular, the dot product must be zero:

$$
(2 - a)(1 - a) = 0
$$

Solving this equation gives:

$$
2 - a = 0 \quad \text{or} \quad 1 - a = 0
$$

$$
a = 2 \quad \text{or} \quad a = 1
$$

Thus, the values of 'a' that satisfy this condition are 2 and 1, which corresponds to Option 4.

**Explanation of Incorrect Options:**

- **Option 1 (-2 and -1):** These values do not satisfy the equation $(2 - a)(1 - a) = 0$. Substituting -2 or -1 into the equation does not yield zero, hence they cannot be correct.

- **Option 2 (-2 and 1):** While 1 is a correct solution, -2 is not. Substituting -2 into the equation does not satisfy the condition for perpendicular vectors.

- **Option 3 (2 and -1):** While 2 is a correct solution, -1 is not. Substituting -1 into the equation does not satisfy the condition for perpendicular vectors.

Therefore, the only correct option is Option 4, which provides the values 2 and 1 for 'a', ensuring that the vectors $\vec{CA}$ and $\vec{CB}$ are perpendicular, forming a right angle at C.