let us define , the length of a vector as `|a| + |b| +|c| `. this definition coincides with the usual definition of the length of a vector `ahati + bhatj + chatk ` 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 condition under which the defined length of a vector \( |a| + |b| + |c| \) coincides with the usual definition of the length of a vector \( \sqrt{a^2 + b^2 + c^2} \).

### Step-by-Step Solution:

1. **Set up the equation:**
We start with the two definitions of the length of a vector:
\[ |a| + |b| + |c| = \sqrt{a^2 + b^2 + c^2} \]

2. **Square both sides:**
To eliminate the square root, we square both sides of the equation:
\[ (|a| + |b| + |c|)^2 = a^2 + b^2 + c^2 \]

3. **Expand the left-hand side:**
Expanding the left-hand side gives:
\[ |a|^2 + |b|^2 + |c|^2 + 2(|a||b| + |b||c| + |c||a|) = a^2 + b^2 + c^2 \]

4. **Recognize that \( |x|^2 = x^2 \):**
Since \( |a|^2 = a^2 \), \( |b|^2 = b^2 \), and \( |c|^2 = c^2 \), we can simplify:
\[ a^2 + b^2 + c^2 + 2(|a||b| + |b||c| + |c||a|) = a^2 + b^2 + c^2 \]

5. **Cancel \( a^2 + b^2 + c^2 \) from both sides:**
This leads to:
\[ 2(|a||b| + |b||c| + |c||a|) = 0 \]

6. **Divide by 2:**
Simplifying gives:
\[ |a||b| + |b||c| + |c||a| = 0 \]

7. **Analyze the implications:**
Since \( |x| \geq 0 \) for any real number \( x \), the only way for the sum \( |a||b| + |b||c| + |c||a| \) to equal zero is if each term is zero. This means:
\[ |a| = 0 \quad \text{or} \quad |b| = 0 \quad \text{or} \quad |c| = 0 \]

8. **Conclude the condition:**
Therefore, at least two of the variables \( a, b, c \) must be zero. This leads us to conclude that:
- Any two of \( a, b, c \) must be zero.

### Final Answer:
The condition under which the defined length of the vector coincides with the usual definition is that any two of \( a, b, c \) must be zero.

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