Three vectors `vec(A) = 2hat(i) - hat(j) + hat(k), vec(B) = hat(i) - 3hat(j) - 5hat(k)`, and `vec(C ) = 3hat(i) - 4hat(j) - 4hat(k)` are sides of an :

Options

A. equilateral triangle
B. right angled triangle
C. isosceles triangle
D. none of these

Answer

To determine the type of triangle formed by the three vectors \( \vec{A} \), \( \vec{B} \), and \( \vec{C} \), we will follow these steps:

### Step 1: Write down the vectors
The vectors are given as:
- \( \vec{A} = 2\hat{i} - \hat{j} + \hat{k} \)
- \( \vec{B} = \hat{i} - 3\hat{j} - 5\hat{k} \)
- \( \vec{C} = 3\hat{i} - 4\hat{j} - 4\hat{k} \)

### Step 2: Calculate the magnitudes of the vectors
The magnitude of a vector \( \vec{V} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by:
\[ |\vec{V}| = \sqrt{a^2 + b^2 + c^2} \]

#### Magnitude of \( \vec{A} \):
\[ |\vec{A}| = \sqrt{(2)^2 + (-1)^2 + (1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \]

#### Magnitude of \( \vec{B} \):
\[ |\vec{B}| = \sqrt{(1)^2 + (-3)^2 + (-5)^2} = \sqrt{1 + 9 + 25} = \sqrt{35} \]

#### Magnitude of \( \vec{C} \):
\[ |\vec{C}| = \sqrt{(3)^2 + (-4)^2 + (-4)^2} = \sqrt{9 + 16 + 16} = \sqrt{41} \]

### Step 3: Compare the magnitudes
Now we have:
- \( |\vec{A}| = \sqrt{6} \)
- \( |\vec{B}| = \sqrt{35} \)
- \( |\vec{C}| = \sqrt{41} \)

Since all three magnitudes are different, this rules out the possibility of an equilateral triangle or an isosceles triangle.

### Step 4: Check for a right-angled triangle
To check if the triangle is a right-angled triangle, we will use the Pythagorean theorem. For a triangle with sides \( a \), \( b \), and \( c \) (where \( c \) is the longest side), the condition is:
\[ c^2 = a^2 + b^2 \]

Assign the sides:
- Let \( a = |\vec{A}| = \sqrt{6} \)
- Let \( b = |\vec{B}| = \sqrt{35} \)
- Let \( c = |\vec{C}| = \sqrt{41} \)

Now, we check:
\[ c^2 = a^2 + b^2 \]
Calculating each side:
\[ c^2 = (\sqrt{41})^2 = 41 \]
\[ a^2 + b^2 = (\sqrt{6})^2 + (\sqrt{35})^2 = 6 + 35 = 41 \]

Since \( c^2 = a^2 + b^2 \), we conclude that the triangle formed by the vectors is a right-angled triangle.

### Conclusion
The triangle formed by the vectors \( \vec{A} \), \( \vec{B} \), and \( \vec{C} \) is a right-angled triangle.

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