The position vectors of the vertices of `Delta ABC` are : `3hat(i)-4hat(j)-4hat(k), 2hat(i)-hat(j)+hat(k)` and `hat(i)-3hat(j)-5hat(k)` respectively.
(a) Find `vec(AB), vec(BC)` and `vec(CA)`
(b) Prove that `Delta ABC` is a right - angles triangle.

Answer

To solve the problem, we will break it down into two parts as given in the question.

### Part (a): Find `vec(AB)`, `vec(BC)`, and `vec(CA)`

1. **Given Position Vectors**:
- Let the position vector of point A be \( \vec{A} = 3\hat{i} - 4\hat{j} - 4\hat{k} \)
- Let the position vector of point B be \( \vec{B} = 2\hat{i} - \hat{j} + \hat{k} \)
- Let the position vector of point C be \( \vec{C} = \hat{i} - 3\hat{j} - 5\hat{k} \)

2. **Finding `vec(AB)`**:
\[ \vec{AB} = \vec{B} - \vec{A} \]
\[ = (2\hat{i} - \hat{j} + \hat{k}) - (3\hat{i} - 4\hat{j} - 4\hat{k}) \]
\[ = (2 - 3)\hat{i} + (-1 + 4)\hat{j} + (1 + 4)\hat{k} \]
\[ = -\hat{i} + 3\hat{j} + 5\hat{k} \]

3. **Finding `vec(BC)`**:
\[ \vec{BC} = \vec{C} - \vec{B} \]
\[ = (\hat{i} - 3\hat{j} - 5\hat{k}) - (2\hat{i} - \hat{j} + \hat{k}) \]
\[ = (1 - 2)\hat{i} + (-3 + 1)\hat{j} + (-5 - 1)\hat{k} \]
\[ = -\hat{i} - 2\hat{j} - 6\hat{k} \]

4. **Finding `vec(CA)`**:
\[ \vec{CA} = \vec{A} - \vec{C} \]
\[ = (3\hat{i} - 4\hat{j} - 4\hat{k}) - (\hat{i} - 3\hat{j} - 5\hat{k}) \]
\[ = (3 - 1)\hat{i} + (-4 + 3)\hat{j} + (-4 + 5)\hat{k} \]
\[ = 2\hat{i} - \hat{j} + \hat{k} \]

### Summary of Part (a):
- \( \vec{AB} = -\hat{i} + 3\hat{j} + 5\hat{k} \)
- \( \vec{BC} = -\hat{i} - 2\hat{j} - 6\hat{k} \)
- \( \vec{CA} = 2\hat{i} - \hat{j} + \hat{k} \)

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### Part (b): Prove that `Delta ABC` is a right-angled triangle

To prove that triangle ABC is a right-angled triangle, we will check if the dot product of any two sides is zero, indicating that they are perpendicular.

1. **Calculate the dot product of `vec(AB)` and `vec(CA)`**:
\[ \vec{AB} \cdot \vec{CA} = (-\hat{i} + 3\hat{j} + 5\hat{k}) \cdot (2\hat{i} - \hat{j} + \hat{k}) \]
\[ = (-1)(2) + (3)(-1) + (5)(1) \]
\[ = -2 - 3 + 5 = 0 \]

Since \( \vec{AB} \cdot \vec{CA} = 0 \), it implies that angle A is \( 90^\circ \).

### Conclusion:
Thus, triangle ABC is a right-angled triangle at vertex A.

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