If `vec(a)` and `vec(b)` are two collinear vectors, then which of the following are incorrect :

Options

A. `vec(b)=lambda vec(a)` for some scalar `lambda`.
B. `vec(a)=pm vec(b)`
C. the respective components of `vec(a)` and `vec(b)` are proportional
D. both the vectors `vec(a)` and `vec(b)` have the same direction, but different magnitude.

Answer

To determine which statements about the collinear vectors \(\vec{a}\) and \(\vec{b}\) are incorrect, we first need to understand the properties of collinear vectors.

### Step-by-Step Solution:

1. **Definition of Collinear Vectors**:
Two vectors \(\vec{a}\) and \(\vec{b}\) are said to be collinear if they lie along the same line. This means that one vector is a scalar multiple of the other. Mathematically, this can be expressed as:
\[ \vec{a} = \lambda \vec{b} \]
where \(\lambda\) is a scalar.

**Hint**: Remember that collinear vectors can point in the same or opposite directions.

2. **Understanding the Implications of Collinearity**:
If \(\vec{a} = \lambda \vec{b}\), then:
- If \(\lambda > 0\), \(\vec{a}\) and \(\vec{b}\) point in the same direction.
- If \(\lambda < 0\), \(\vec{a}\) and \(\vec{b}\) point in opposite directions.
- If \(\lambda = 0\), then \(\vec{a}\) is the zero vector.

**Hint**: Think about how the scalar \(\lambda\) affects the direction of the vectors.

3. **Vector Components**:
If we express the vectors in terms of their components:
\[ \vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \]
\[ \vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \]
Then, for \(\vec{a}\) to be collinear with \(\vec{b}\), the components must satisfy:
\[ a_1 = \lambda b_1, \quad a_2 = \lambda b_2, \quad a_3 = \lambda b_3 \]
This means that the components of the vectors are proportional.

**Hint**: Check if the ratios of corresponding components are equal.

4. **Same Direction but Different Magnitudes**:
If both vectors have the same direction but different magnitudes, they can still be collinear. For example, if \(\vec{a} = 2\vec{b}\), they are collinear because they point in the same direction.

**Hint**: Consider examples of vectors with different lengths but the same direction.

5. **Conclusion about Incorrect Statements**:
- If a statement claims that two collinear vectors must have the same magnitude, it is incorrect. Collinear vectors can have different magnitudes as long as they point in the same or opposite directions.
- Therefore, any statement that implies collinear vectors cannot have different magnitudes is incorrect.

**Hint**: Look for statements that contradict the properties of collinear vectors.

### Final Answer:
Based on the analysis, the incorrect statement regarding collinear vectors is the one that suggests they cannot have different magnitudes while still being collinear.