If `vec(P) xx vec(Q) =vec(R )`, then which of the following statements is not true?

Options

A. `vec(R ) _|_ vec(P)`
B. `vec(R ) _|_ vec(Q)`
C. `vec(R ) _|_ (vec(P) + vec(Q))`
D. `vec(R ) _|_ (vec(P) xx vec(Q))`

Answer

To solve the question, we need to analyze the properties of the cross product of two vectors \( \vec{P} \) and \( \vec{Q} \) which results in a vector \( \vec{R} \). The question asks us to identify which statement about these vectors is not true.

### Step-by-step Solution:

1. **Understanding the Cross Product**:
The cross product \( \vec{P} \times \vec{Q} \) results in a vector \( \vec{R} \) that is perpendicular to both \( \vec{P} \) and \( \vec{Q} \). This is a fundamental property of the cross product.

2. **Analyzing the Statements**:
We need to evaluate the given statements to find out which one is false. The statements typically involve relationships between \( \vec{R} \) and the vectors \( \vec{P} \) and \( \vec{Q} \).

3. **Checking Each Statement**:
- **Statement 1**: \( \vec{R} \) is perpendicular to \( \vec{P} \).
- This statement is true because \( \vec{R} \) is defined as \( \vec{P} \times \vec{Q} \), which is perpendicular to \( \vec{P} \).

- **Statement 2**: \( \vec{R} \) is perpendicular to \( \vec{Q} \).
- This statement is also true for the same reason as above; \( \vec{R} \) is perpendicular to \( \vec{Q} \).

- **Statement 3**: \( \vec{R} \) is perpendicular to \( \vec{P} + \vec{Q} \).
- This statement is true as well. Since \( \vec{R} \) is perpendicular to both \( \vec{P} \) and \( \vec{Q} \), it is also perpendicular to any linear combination of these vectors, including \( \vec{P} + \vec{Q} \).

- **Statement 4**: \( \vec{R} \) is perpendicular to itself.
- This statement is false. A vector cannot be perpendicular to itself. Therefore, this is the statement that is not true.

4. **Conclusion**:
The statement that is not true is that \( \vec{R} \) is perpendicular to itself.

### Final Answer:
The statement that is not true is: **\( \vec{R} \) is perpendicular to itself.**