Question:
Define the length of \( a\hat{i} + b\hat{j} + c\hat{k} \) as \( |a| + |b| + |c| \). This definition coincides with the usual definition if and only if
Define the length of \( a\hat{i} + b\hat{j} + c\hat{k} \) as \( |a| + |b| + |c| \). This definition coincides with the usual definition if and only if
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Define the length of $a\hati+b\hatj+c\hatk$ as $|a|+|b|+|c|$. This definition coincides with the usual definition iff
Updated On: Apr 15, 2026
- $a=b=c=0$
- any two of a, b and c are zero
- any one of a, b and c is zero
- $a+b+c=0$
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The Correct Option is B
Solution and Explanation
Step 1: Concept
Equate $\sqrt{a^2+b^2+c^2} = |a|+|b|+|c|$.
Step 2: Analysis
Squaring both sides: $a^2+b^2+c^2 = a^2+b^2+c^2 + 2(|ab|+|bc|+|ca|)$.
Step 3: Evaluation
This implies $|ab|+|bc|+|ca| = 0$. For this to hold, $ab=0, bc=0,$ and $ca=0$.
Step 4: Conclusion
This condition is only satisfied if at least two of the variables $a, b, c$ are zero.
Final Answer: (b)
Equate $\sqrt{a^2+b^2+c^2} = |a|+|b|+|c|$.
Step 2: Analysis
Squaring both sides: $a^2+b^2+c^2 = a^2+b^2+c^2 + 2(|ab|+|bc|+|ca|)$.
Step 3: Evaluation
This implies $|ab|+|bc|+|ca| = 0$. For this to hold, $ab=0, bc=0,$ and $ca=0$.
Step 4: Conclusion
This condition is only satisfied if at least two of the variables $a, b, c$ are zero.
Final Answer: (b)
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