Consider the following statements:
1. The magnitude of `vec(a) xx vec(b)` is same as the area of a triangle with sides `vec(a) and vec(b)`.
2. If `vec(a) xx vec(b)=vec(0)` where `vec(a) ne 0. vec(b) ne 0` then `vec(a) = lamda vec(b)` which of the following statements is/are correct?

Options

A. A) 1 only
B. B) 2 only
C. C) Both 1 and 2
D. D) Neither 1 nor 2

Answer

To determine the correctness of the two statements regarding vectors \(\vec{a}\) and \(\vec{b}\), let's analyze each statement step by step.

### Step 1: Analyze Statement 1
**Statement 1**: The magnitude of \(\vec{a} \times \vec{b}\) is the same as the area of a triangle with sides \(\vec{a}\) and \(\vec{b}\).

1. The cross product \(\vec{a} \times \vec{b}\) gives a vector whose magnitude is equal to the area of the parallelogram formed by the vectors \(\vec{a}\) and \(\vec{b}\).
2. The area of a triangle formed by the same vectors is half the area of the parallelogram.
3. Therefore, the magnitude of \(\vec{a} \times \vec{b}\) is not equal to the area of the triangle, but rather twice that area.

**Conclusion**: Statement 1 is **incorrect**.

### Step 2: Analyze Statement 2
**Statement 2**: If \(\vec{a} \times \vec{b} = \vec{0}\) where \(\vec{a} \neq \vec{0}\) and \(\vec{b} \neq \vec{0}\), then \(\vec{a} = \lambda \vec{b}\).

1. The cross product \(\vec{a} \times \vec{b} = \vec{0}\) indicates that the vectors \(\vec{a}\) and \(\vec{b}\) are parallel.
2. For two non-zero vectors, the condition \(\vec{a} \times \vec{b} = \vec{0}\) implies that the sine of the angle \(\theta\) between them is zero, which occurs when \(\theta = 0^\circ\) or \(\theta = 180^\circ\).
3. This means that \(\vec{a}\) can be expressed as a scalar multiple of \(\vec{b}\), i.e., \(\vec{a} = \lambda \vec{b}\) for some scalar \(\lambda\).

**Conclusion**: Statement 2 is **correct**.

### Final Conclusion
- Statement 1 is incorrect.
- Statement 2 is correct.

Thus, the correct answer is that only Statement 2 is true.

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