Question

In \(\triangle ABC\), if \[ \overrightarrow{AB}=2\hat{i}-\hat{j}+2\hat{k} \] and \[ \overrightarrow{AC}=3\hat{i}-3\hat{j}+4\hat{k}, \] then the triangle \(ABC\) is

• A. an equilateral triangle
• B. a right angled triangle
• C. an isosceles triangle
• D. a scalene triangle \bigskip

Hint:

When side vectors of a triangle are given, first find their magnitudes. If the third side is required, use \[ \overrightarrow{BC} = \overrightarrow{AC}-\overrightarrow{AB}. \] Comparing the side lengths immediately identifies whether the triangle is equilateral, isosceles, or scalene.

Answer 1

The Correct Option is C

Concept: Find the side lengths using the distance formula on the given vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\), then compute \(\overrightarrow{BC}\) and its length to compare.

Step 1: Compute lengths \(AB\) and \(AC\). \[ AB = |\overrightarrow{AB}| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4+1+4} = \sqrt{9} = 3. \] \[ AC = |\overrightarrow{AC}| = \sqrt{3^2 + (-3)^2 + 4^2} = \sqrt{9+9+16} = \sqrt{34}. \]

Step 2: Compute \(\overrightarrow{BC}\) and its length. \[ \overrightarrow{BC} = \overrightarrow{AC} - \overrightarrow{AB} = (3-2)\hat{i} + (-3-(-1))\hat{j} + (4-2)\hat{k} = \hat{i} - 2\hat{j} + 2\hat{k}. \] \[ BC = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{1+4+4} = \sqrt{9} = 3. \]

Step 3: Compare sides and conclude. \(AB = 3, BC = 3, AC = \sqrt{34}\). Two sides equal implies the triangle is isosceles.

Step 4: Write the final answer. \[ \boxed{\text{An isosceles triangle}} \]