Question

The position vectors of the vertices of a triangle are \( 4\hat{i} + 5\hat{j} + \hat{k} \), \( 2\hat{i} + 4\hat{j} - \hat{k} \) and \( 3\hat{i} + 6\hat{j} - 3\hat{k} \). Then the triangle is

• A. Right-angled but not isosceles
• B. Isosceles but not right-angled
• C. Right-angled isosceles
• D. Equilateral

Hint:

Use the dot product to check for right angles between vectors. If the dot product is zero, the vectors are perpendicular, and the triangle is right-angled.

Answer 1

The Correct Option is A

Step 1: Find the vectors representing the sides of the triangle.
The position vectors are given for the vertices \( A(4, 5, 1) \), \( B(2, 4, -1) \), and \( C(3, 6, -3) \). The vectors representing the sides are: \[ \overrightarrow{AB} = B - A = (2 - 4, 4 - 5, -1 - 1) = (-2, -1, -2) \] \[ \overrightarrow{AC} = C - A = (3 - 4, 6 - 5, -3 - 1) = (-1, 1, -4) \] \[ \overrightarrow{BC} = C - B = (3 - 2, 6 - 4, -3 - (-1)) = (1, 2, -2) \]
Step 2: Check for right angles using the dot product.
To check if the triangle is right-angled, calculate the dot product of two sides. For \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \): \[ \overrightarrow{AB} \cdot \overrightarrow{AC} = (-2)(-1) + (-1)(1) + (-2)(-4) = 2 - 1 + 8 = 9 \neq 0 \] For \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \): \[ \overrightarrow{AB} \cdot \overrightarrow{BC} = (-2)(1) + (-1)(2) + (-2)(-2) = -2 - 2 + 4 = 0 \] Since the dot product is zero, the angle between \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \) is \( 90^\circ \), so the triangle is right-angled.
Step 3: Conclusion.
The triangle is right-angled but not isosceles. The correct answer is (1) Right-angled but not isosceles.