The three vectors `7i-11j+k, 5i+3j-2k` and `12i-8j-k` form

Options

A. an equilateral `Delta`
B. rt. Angled `Delta`
C. isosceles `Delta`
D. collinear vectors

Answer

To determine the type of triangle formed by the three vectors \( \mathbf{A} = 7\mathbf{i} - 11\mathbf{j} + \mathbf{k} \), \( \mathbf{B} = 5\mathbf{i} + 3\mathbf{j} - 2\mathbf{k} \), and \( \mathbf{C} = 12\mathbf{i} - 8\mathbf{j} - \mathbf{k} \), we will calculate the magnitudes of each vector and check the conditions for different types of triangles.

### Step 1: Calculate the Magnitude of Vector A
The magnitude of vector \( \mathbf{A} \) is calculated using the formula:
\[ |\mathbf{A}| = \sqrt{(x^2 + y^2 + z^2)} \]
For \( \mathbf{A} = 7\mathbf{i} - 11\mathbf{j} + \mathbf{k} \):
\[ |\mathbf{A}| = \sqrt{(7^2 + (-11)^2 + 1^2)} = \sqrt{(49 + 121 + 1)} = \sqrt{171} \]

### Step 2: Calculate the Magnitude of Vector B
For \( \mathbf{B} = 5\mathbf{i} + 3\mathbf{j} - 2\mathbf{k} \):
\[ |\mathbf{B}| = \sqrt{(5^2 + 3^2 + (-2)^2)} = \sqrt{(25 + 9 + 4)} = \sqrt{38} \]

### Step 3: Calculate the Magnitude of Vector C
For \( \mathbf{C} = 12\mathbf{i} - 8\mathbf{j} - \mathbf{k} \):
\[ |\mathbf{C}| = \sqrt{(12^2 + (-8)^2 + (-1)^2)} = \sqrt{(144 + 64 + 1)} = \sqrt{209} \]

### Step 4: Check for Triangle Type
To determine the type of triangle formed by these vectors, we need to check if any two sides are equal (for isosceles), if all three sides are equal (for equilateral), or if the Pythagorean theorem holds (for right-angled).

1. **Check for Isosceles or Equilateral**:
- \( |\mathbf{A}| = \sqrt{171} \)
- \( |\mathbf{B}| = \sqrt{38} \)
- \( |\mathbf{C}| = \sqrt{209} \)

Since \( |\mathbf{A}| \), \( |\mathbf{B}| \), and \( |\mathbf{C}| \) are all different, the triangle is neither equilateral nor isosceles.

2. **Check for Right-Angled Triangle**:
We check if \( |\mathbf{A}|^2 + |\mathbf{B}|^2 = |\mathbf{C}|^2 \):
\[ |\mathbf{A}|^2 = 171, \quad |\mathbf{B}|^2 = 38, \quad |\mathbf{C}|^2 = 209 \]
\[ 171 + 38 = 209 \]
This satisfies the Pythagorean theorem, indicating that the triangle is a right-angled triangle.

### Conclusion
The three vectors \( 7\mathbf{i} - 11\mathbf{j} + \mathbf{k} \), \( 5\mathbf{i} + 3\mathbf{j} - 2\mathbf{k} \), and \( 12\mathbf{i} - 8\mathbf{j} - \mathbf{k} \) form a right-angled triangle.