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: For vectors, \[ |\vec{a}|=\sqrt{a_1^2+a_2^2+a_3^2} \] gives the length of a side. If any two sides of a triangle are equal, then the triangle is isosceles. Also, \[ \overrightarrow{BC} = \overrightarrow{AC}-\overrightarrow{AB}. \]

Step 1: Find the lengths of \(AB\) and \(AC\). Given \[ \overrightarrow{AB}=2\hat{i}-\hat{j}+2\hat{k}. \] Hence, \[ AB = \sqrt{2^2+(-1)^2+2^2} = \sqrt{9} = 3. \] Also, \[ \overrightarrow{AC}=3\hat{i}-3\hat{j}+4\hat{k}. \] Therefore, \[ AC = \sqrt{3^2+(-3)^2+4^2} = \sqrt{34}. \]

Step 2: Find the vector \(\overrightarrow{BC}\). \[ \overrightarrow{BC} = \overrightarrow{AC}-\overrightarrow{AB}. \] \[ = (3\hat{i}-3\hat{j}+4\hat{k}) -(2\hat{i}-\hat{j}+2\hat{k}). \] \[ = \hat{i}-2\hat{j}+2\hat{k}. \] Hence, \[ BC = \sqrt{1^2+(-2)^2+2^2} = \sqrt{9} = 3. \]

Step 3: Compare the side lengths. We have \[ AB=3, \] \[ BC=3, \] and \[ AC=\sqrt{34}. \] Thus, \[ AB=BC. \] Therefore, two sides of the triangle are equal.

Step 4: Write the final answer. Hence, \(\triangle ABC\) is an isosceles triangle. \[ \boxed{\text{An isosceles triangle}} \]