Which of the following quantities are independent of the choice of orientation of the co-ordinate axes?

Options

A. `a+b`
B. `a_(x)+b_(y)`
C. `|a+b-c|`
D. angle between `a` and `b`

Answer

To determine which quantities are independent of the choice of orientation of the coordinate axes, we need to analyze the nature of various quantities in vector mathematics.

### Step-by-Step Solution:

1. **Understanding Vectors**:
- A vector has both magnitude and direction. The components of a vector depend on the orientation of the coordinate axes.

2. **Magnitude of a Vector**:
- The magnitude of a vector is calculated using the formula \( | \vec{A} | = \sqrt{A_x^2 + A_y^2 + A_z^2} \). This value is independent of the orientation of the axes because it is a scalar quantity derived from the lengths of the components, not their directions.

3. **Sum of Vectors**:
- When adding vectors (e.g., \( \vec{A} + \vec{B} \)), the resultant vector's magnitude and direction can change with the orientation of the axes. However, the resultant vector itself, when considered as a whole, is independent of the specific orientation of the axes as long as we are consistent in our calculations.

4. **Angle Between Two Vectors**:
- The angle between two vectors is determined using the dot product formula:
\[ \cos(\theta) = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|} \]
This angle is also independent of the orientation of the coordinate axes.

5. **Scalar Multiplication of a Vector**:
- A scalar multiplied by a vector (e.g., \( \lambda \vec{A} \)) results in a vector that maintains its direction relative to the original vector. The scalar does not depend on the orientation of the axes.

6. **Component-Based Quantities**:
- Quantities expressed in terms of components along the axes (like \( 2A_x + 3A_y \)) depend on the orientation of the axes. Changing the orientation would change the values of \( A_x \) and \( A_y \).

### Conclusion:
The quantities that are independent of the choice of orientation of the coordinate axes are:
- The magnitude of a vector.
- The angle between two vectors.
- The resultant vector from vector addition (as a whole).
- Scalar multiplication of a vector.