If a and b are two non-zero and non-collinear vectors then a+b and a-b are

Options

A. linearly dependent vectors
B. linearly independent vectors
C. linearly dependent annd independent vectors
D. none of these

Explanation

To solve the problem, we need to determine the relationship between the vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) given that \( \mathbf{a} \) and \( \mathbf{b} \) are two non-zero and non-collinear vectors.

### Step-by-step Solution:

1. **Understanding Non-collinearity**:
Since \( \mathbf{a} \) and \( \mathbf{b} \) are non-collinear, it means that they do not lie on the same line. This implies that there are no scalar multiples \( k \) such that \( \mathbf{a} = k \mathbf{b} \) for any scalar \( k \).

2. **Linear Independence of \( \mathbf{a} \) and \( \mathbf{b} \)**:
The fact that \( \mathbf{a} \) and \( \mathbf{b} \) are non-zero and non-collinear means they are linearly independent. This can be expressed mathematically as:
\[ x_1 \mathbf{a} + x_2 \mathbf{b} = \mathbf{0} \implies x_1 = 0 \text{ and } x_2 = 0 \]
for any scalars \( x_1 \) and \( x_2 \).

3. **Expressing \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \)**:
Now, we consider the vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \). We want to check if these two vectors are linearly independent.

4. **Setting Up the Equation**:
We set up the equation:
\[ x_1 (\mathbf{a} + \mathbf{b}) + x_2 (\mathbf{a} - \mathbf{b}) = \mathbf{0} \]
Expanding this gives:
\[ x_1 \mathbf{a} + x_1 \mathbf{b} + x_2 \mathbf{a} - x_2 \mathbf{b} = \mathbf{0} \]
This simplifies to:
\[ (x_1 + x_2) \mathbf{a} + (x_1 - x_2) \mathbf{b} = \mathbf{0} \]

5. **Analyzing the Coefficients**:
For the above equation to hold true, both coefficients must equal zero:
\[ x_1 + x_2 = 0 \quad \text{(1)} \]
\[ x_1 - x_2 = 0 \quad \text{(2)} \]

6. **Solving the System of Equations**:
From equation (2), we have \( x_1 = x_2 \). Substituting this into equation (1):
\[ x_1 + x_1 = 0 \implies 2x_1 = 0 \implies x_1 = 0 \]
Thus, \( x_2 = 0 \) as well.

7. **Conclusion**:
Since the only solution to the equation is \( x_1 = 0 \) and \( x_2 = 0 \), the vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) are linearly independent.

### Final Answer:
The vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) are linearly independent.

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