Question

If \(4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}\) and \(2\hat{i}+5\hat{j}+7\hat{k}\) are the position vectors of the vertices A, B and C respectively of the triangle ABC, then the largest angle of the triangle is (are) ...

• A. A
• B. C
• C. A and C
• D. B and C

Hint:

Find the three side lengths; the largest angle is opposite the longest side.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
The largest angle of a triangle lies opposite its longest side.

Step 2: Find the sides
With \(A(4,7,8),B(2,3,4),C(2,5,7)\):
\(AB=\sqrt{4+16+16}=6\), \(BC=\sqrt{0+4+9}=\sqrt{13}\), \(AC=\sqrt{4+4+1}=3\).

Step 3: Identify
The longest side is \(AB=6\), which is opposite vertex \(C\).

Step 4: Verify
\(AB^{2}=36>BC^{2}+AC^{2}=22\), so the angle at \(C\) is obtuse.

Final Answer:
The largest angle is at vertex C. \[ \boxed{\text{(B) }\text{C}} \]