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 solve the problem, we need to analyze the properties of collinear vectors. Two vectors are said to be collinear if they lie along the same line, which means one vector can be expressed as a scalar multiple of the other.

Let's denote the vectors as:
- \(\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}\)
- \(\vec{b} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k}\)

Now, we will evaluate each option provided in the question to determine which one is incorrect.

### Step 1: Evaluate Option 1
**Option 1:** \(\vec{b} = \lambda \vec{a}\) for some scalar \(\lambda\).

- This statement is true because if \(\vec{a}\) and \(\vec{b}\) are collinear, then one vector can be expressed as a scalar multiple of the other.

### Step 2: Evaluate Option 2
**Option 2:** \(\vec{a} = \pm \vec{b}\).

- This statement is also true. If \(\vec{a}\) is equal to \(\vec{b}\), they point in the same direction. If \(\vec{a} = -\vec{b}\), they point in opposite directions. Both cases indicate collinearity.

### Step 3: Evaluate Option 3
**Option 3:** The respective components of \(\vec{a}\) and \(\vec{b}\) are proportional.

- This statement is true. For collinear vectors, the ratios of their corresponding components must be equal, i.e., \(\frac{a_1}{b_1} = \frac{a_2}{b_2} = \frac{a_3}{b_3} = \lambda\).

### Step 4: Evaluate Option 4
**Option 4:** Both vectors \(\vec{a}\) and \(\vec{b}\) have the same direction but different magnitudes.

- This statement is incorrect. While collinear vectors can have the same direction, they can also have opposite directions, and their magnitudes can differ. However, they cannot have the same direction and different magnitudes simultaneously; if they have the same direction, they must be scalar multiples of each other with a positive scalar.

### Conclusion
Thus, the incorrect option is Option 4.

### Final Answer
**Option 4 is incorrect.**

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