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 whether the vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) are linearly dependent or independent given that \( \mathbf{a} \) and \( \mathbf{b} \) are non-zero and non-collinear vectors.

### Step-by-Step Solution:

1. **Understanding Non-Collinearity**:
- Since \( \mathbf{a} \) and \( \mathbf{b} \) are non-collinear, they do not lie along the same line. This means that they cannot be expressed as scalar multiples of each other.

**Hint**: Remember that two vectors are collinear if one is a scalar multiple of the other.

2. **Defining the Vectors**:
- We denote the vectors as \( \mathbf{a} \) and \( \mathbf{b} \). The vectors \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) can be visualized as the resultant vectors when you add or subtract \( \mathbf{b} \) from \( \mathbf{a} \).

**Hint**: Visualizing vectors can help understand their relationships better.

3. **Finding Linear Dependence or Independence**:
- Two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are linearly dependent if there exist scalars \( k_1 \) and \( k_2 \), not both zero, such that:
\[ k_1 \mathbf{u} + k_2 \mathbf{v} = \mathbf{0} \]
- For \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) to be dependent, we would need to find scalars \( k_1 \) and \( k_2 \) such that:
\[ k_1 (\mathbf{a} + \mathbf{b}) + k_2 (\mathbf{a} - \mathbf{b}) = \mathbf{0} \]

**Hint**: Set up the equation for linear dependence and see if you can find non-trivial solutions.

4. **Setting Up the Equation**:
- Expanding the equation gives:
\[ k_1 \mathbf{a} + k_1 \mathbf{b} + k_2 \mathbf{a} - k_2 \mathbf{b} = \mathbf{0} \]
- This simplifies to:
\[ (k_1 + k_2) \mathbf{a} + (k_1 - k_2) \mathbf{b} = \mathbf{0} \]

**Hint**: For this equation to hold true, both coefficients of \( \mathbf{a} \) and \( \mathbf{b} \) must equal zero.

5. **Analyzing the Coefficients**:
- From the equation:
\[ k_1 + k_2 = 0 \quad \text{(1)} \]
\[ k_1 - k_2 = 0 \quad \text{(2)} \]
- Solving these equations gives:
- From (2), \( k_1 = k_2 \).
- Substituting into (1) gives \( k_1 + k_1 = 0 \) or \( 2k_1 = 0 \), hence \( k_1 = 0 \) and \( k_2 = 0 \).

**Hint**: If the only solution to the equations is the trivial solution (both scalars are zero), then the vectors are independent.

6. **Conclusion**:
- Since the only solution is \( k_1 = 0 \) and \( k_2 = 0 \), we conclude that \( \mathbf{a} + \mathbf{b} \) and \( \mathbf{a} - \mathbf{b} \) are linearly independent vectors.

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