Which of the following vector identies is false ?

Explanation

To determine which vector identity is false, we will analyze each of the provided vector identities step by step.

### Step 1: Analyze Identity 1
**Identity 1:** \( \mathbf{P} + \mathbf{Q} = \mathbf{Q} + \mathbf{P} \)

**Explanation:** This identity states that the sum of two vectors is commutative. This is a fundamental property of vector addition.

**Conclusion:** This identity is **true**.

### Step 2: Analyze Identity 2
**Identity 2:** \( \mathbf{P} + \mathbf{Q} = \mathbf{Q} \times \mathbf{P} \)

**Explanation:** This identity states that the sum of two vectors is equal to the cross product of those vectors. However, the sum of two vectors (a vector) and the cross product of two vectors (which results in another vector) are fundamentally different operations. Therefore, this identity cannot hold true.

**Conclusion:** This identity is **false**.

### Step 3: Analyze Identity 3
**Identity 3:** \( \mathbf{P} \cdot \mathbf{Q} = \mathbf{Q} \cdot \mathbf{P} \)

**Explanation:** This identity states that the dot product of two vectors is commutative. This is a well-known property of the dot product.

**Conclusion:** This identity is **true**.

### Step 4: Analyze Identity 4
**Identity 4:** \( \mathbf{P} \times \mathbf{Q} \neq \mathbf{Q} \times \mathbf{P} \)

**Explanation:** This identity states that the cross product of two vectors is not equal when the order of the vectors is reversed. This is also a fundamental property of the cross product, which states that \( \mathbf{P} \times \mathbf{Q} = -(\mathbf{Q} \times \mathbf{P}) \).

**Conclusion:** This identity is **true**.

### Final Conclusion
After analyzing all four identities, we find that the **false identity** is:

**Identity 2:** \( \mathbf{P} + \mathbf{Q} = \mathbf{Q} \times \mathbf{P} \)

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