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 C

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.

Answer 2

To classify this triangle, we need both the three side lengths (to check for equal sides) and whether any angle is exactly \( 90^\circ \) (to check for a right angle). Using the position vectors \( A=(4,5,1) \), \( B=(2,4,-1) \), \( C=(3,6,-3) \):

\[ \overrightarrow{AB}=(-2,-1,-2),\quad |\overrightarrow{AB}|^2=4+1+4=9 \] \[ \overrightarrow{BC}=(1,2,-2),\quad |\overrightarrow{BC}|^2=1+4+4=9 \] \[ \overrightarrow{AC}=(-1,1,-4),\quad |\overrightarrow{AC}|^2=1+1+16=18 \]

So \( |AB|=|BC|=3 \) and \( |AC|=3\sqrt{2} \), meaning two sides are equal. Checking the angle at \( B \): \( \overrightarrow{AB}\cdot\overrightarrow{BC}=(-2)(1)+(-1)(2)+(-2)(-2)=-2-2+4=0 \), so the angle at \( B \) is exactly \( 90^\circ \).

1. Right-angled but not isosceles: We do have a right angle at \( B \), but we also found \( |AB|=|BC|=3 \), so the triangle IS isosceles as well, ruling this option out.
2. Isosceles but not right-angled: We do have two equal sides, but we also confirmed \( \overrightarrow{AB}\cdot\overrightarrow{BC}=0 \), meaning there is a right angle, so this option is incomplete too.
3. Right-angled isosceles: Both conditions are satisfied together, two sides of length \( 3 \) each and a right angle between them at \( B \), so this describes the triangle fully and correctly.
4. Equilateral: All three sides would need to be equal, but \( |AC|=3\sqrt2 \) is different from \( |AB|=|BC|=3 \), so the triangle is not equilateral.

The side-length and dot-product checks together confirm the triangle has both a right angle and two equal sides.

Therefore, the correct answer is Right-angled isosceles.