Question

In a triangle \(ABC\), if \[ \overrightarrow{AB}=\hat{i}-2\hat{j}+3\hat{k}, \qquad \overrightarrow{BC}=3\hat{i}+2\hat{j}-2\hat{k}, \] then the triangle is

• A. obtuse angled triangle
• B. isosceles triangle
• C. isosceles right angled triangle
• D. equilateral triangle

Hint:

For vectors representing the sides of a triangle, \[ \overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}. \] Compare the magnitudes of the three sides to identify the type of triangle.

Answer 1

The Correct Option is B

Step 1: Find the lengths of the given sides. The magnitude of \(\overrightarrow{AB}\) is \[ |\overrightarrow{AB}| = \sqrt{1^2+(-2)^2+3^2} = \sqrt{14}. \] Similarly, \[ |\overrightarrow{BC}| = \sqrt{3^2+2^2+(-2)^2} = \sqrt{17}. \]

Step 2: Find the third side. Since \[ \overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}, \] we obtain \[ \overrightarrow{AC} = (1+3)\hat{i}+(-2+2)\hat{j}+(3-2)\hat{k} = 4\hat{i}+\hat{k}. \] Hence, \[ |\overrightarrow{AC}| = \sqrt{4^2+0^2+1^2} = \sqrt{17}. \]

Step 3: Identify the type of triangle. Since \[ |\overrightarrow{BC}| = |\overrightarrow{AC}| = \sqrt{17}, \] two sides are equal. Therefore, the triangle is an \[ \boxed{\text{isosceles triangle}.} \] Hence, the correct option is \(\boxed{(B)}\).