Question

The three vector \(\vec{A}=3 \hat{i}-2 \hat{j}+\hat{k}, \vec{B}=\hat{i}-3 \hat{j}+5 k\) and \(\vec{C}=2 \hat{i}-\hat{j}+4 \hat{k}\) will form

• A. isosceles triangle.
• B. equilateral triangle.
• C. no triangle.
• D. right angled triangle.

Hint:

Three vectors form a triangle only if one of them equals the sum of the other two (up to sign).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Vectors form a closed triangle only if they add to zero in some combination, for example \(\vec A\pm\vec B\pm\vec C=\vec 0\).

Step 2: Test the combinations
\(\vec A+\vec B=4\hat i-5\hat j+6\hat k\ne\pm\vec C\)
\(\vec A-\vec B=2\hat i+\hat j-4\hat k\ne\pm\vec C\)
\(\vec A\pm\vec C\) gives \(5\hat i-3\hat j+5\hat k\) or \(\hat i-\hat j-3\hat k\), neither equal to \(\pm\vec B\).

Step 3: Conclusion
No combination gives the zero vector, so the three vectors cannot form a triangle: option (C).

Final Answer:
The three vectors do not form any triangle. \[ \boxed{\text{(C) }\text{no triangle}} \]