Question

If \( \vec{i} - 2\vec{j} + 3\vec{k}, 2\vec{i} + 3\vec{j} - \vec{k}, -3\vec{i} - \vec{j} - 2\vec{k} \) are the position vectors of three points A, B, C respectively, then A, B, C:

• A. are collinear points
• B. form an isosceles triangle which is not equilateral
• C. form an equilateral triangle
• D. form a scalene triangle

Hint:

To check whether three points are collinear or form a specific type of triangle, use the distance formula and vector operations like cross product to verify the conditions.

Answer 1

The Correct Option is C

We are given the position vectors of points \( A, B, C \): \[ \vec{A} = \hat{i} - 2\hat{j} + 3\hat{k}, \quad \vec{B} = 2\hat{i} + 3\hat{j} - \hat{k}, \quad \vec{C} = -3\hat{i} - \hat{j} - 2\hat{k}. \] To determine the type of triangle formed by points A, B, and C, we calculate the distances \( AB \), \( BC \), and \( CA \) using the distance formula between two points in 3D space: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}. \] After computing the distances, we find that all the distances are equal, thus forming an equilateral triangle. Thus, the correct answer is that A, B, and C form an equilateral triangle.

Answer 2

Given points:
\[ \vec{A} = \vec{i} - 2\vec{j} + 3\vec{k}, \quad \vec{B} = 2\vec{i} + 3\vec{j} - \vec{k}, \quad \vec{C} = -3\vec{i} - \vec{j} - 2\vec{k} \] Check if A, B, and C form an equilateral triangle.

Step 1: Calculate the vectors representing sides:
\[ \vec{AB} = \vec{B} - \vec{A} = (2 - 1)\vec{i} + (3 - (-2))\vec{j} + (-1 - 3)\vec{k} = \vec{i} + 5\vec{j} - 4\vec{k} \] \[ \vec{BC} = \vec{C} - \vec{B} = (-3 - 2)\vec{i} + (-1 - 3)\vec{j} + (-2 - (-1))\vec{k} = -5\vec{i} - 4\vec{j} - \vec{k} \] \[ \vec{CA} = \vec{A} - \vec{C} = (1 - (-3))\vec{i} + (-2 - (-1))\vec{j} + (3 - (-2))\vec{k} = 4\vec{i} - \vec{j} + 5\vec{k} \]

Step 2: Calculate the lengths of each side:
\[ |\vec{AB}| = \sqrt{1^2 + 5^2 + (-4)^2} = \sqrt{1 + 25 + 16} = \sqrt{42} \] \[ |\vec{BC}| = \sqrt{(-5)^2 + (-4)^2 + (-1)^2} = \sqrt{25 + 16 + 1} = \sqrt{42} \] \[ |\vec{CA}| = \sqrt{4^2 + (-1)^2 + 5^2} = \sqrt{16 + 1 + 25} = \sqrt{42} \]

Step 3: Since all three sides have equal length \( \sqrt{42} \), the triangle formed by points A, B, and C is equilateral.

Therefore,
\[ \boxed{\text{A, B, C form an equilateral triangle}} \]