Let us define the length of a vector `ahati+bhatj+chatk` and `|a|+|b|+|c|`. This definition coincides with the usual definition of length of a vector `ahati+bhatj+chatk` if an only if

Options

A. a=b=c=0
B. any two of a,b and c are zero
C. any one of a,b and c is zero
D. a+b+c=0

Explanation

To solve the problem, we need to find the conditions under which the defined length of a vector \( \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} \) given by \( |a| + |b| + |c| \) coincides with the usual definition of length \( |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \).

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- The usual length of the vector \( \vec{v} \) is given by:
\[ |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \]
- The defined length of the vector is:
\[ L = |a| + |b| + |c| \]

2. **Setting Up the Condition**:
- We want to find when:
\[ |a| + |b| + |c| = \sqrt{a^2 + b^2 + c^2} \]

3. **Squaring Both Sides**:
- Squaring both sides gives:
\[ (|a| + |b| + |c|)^2 = a^2 + b^2 + c^2 \]
- Expanding the left side:
\[ |a|^2 + |b|^2 + |c|^2 + 2(|a||b| + |b||c| + |c||a|) = a^2 + b^2 + c^2 \]
- Since \( |a|^2 = a^2 \), \( |b|^2 = b^2 \), and \( |c|^2 = c^2 \), we can simplify:
\[ a^2 + b^2 + c^2 + 2(|a||b| + |b||c| + |c||a|) = a^2 + b^2 + c^2 \]

4. **Cancelling Terms**:
- Cancelling \( a^2 + b^2 + c^2 \) from both sides results in:
\[ 2(|a||b| + |b||c| + |c||a|) = 0 \]

5. **Analyzing the Result**:
- The equation \( |a||b| + |b||c| + |c||a| = 0 \) implies that each term must be zero because \( |a|, |b|, |c| \) are non-negative.
- Therefore, at least two of \( |a|, |b|, |c| \) must be zero.

6. **Conclusion**:
- Hence, the condition for the defined length to coincide with the usual length is that at least two of \( a, b, c \) must be zero.

### Final Answer:
The definition coincides with the usual definition of length if and only if any two of \( a, b, c \) are zero.