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 problem, we need to analyze the given vector equation and the properties of the cross product.

### Step-by-Step Solution:

1. **Understanding the Cross Product**:
- The cross product of two vectors \(\vec{P}\) and \(\vec{Q}\) is given by \(\vec{P} \times \vec{Q} = \vec{R}\).
- The resulting vector \(\vec{R}\) is perpendicular to the plane formed by the vectors \(\vec{P}\) and \(\vec{Q}\).

2. **Identifying Perpendicularity**:
- Since \(\vec{R}\) is the result of the cross product, it is perpendicular to both \(\vec{P}\) and \(\vec{Q}\).
- Therefore, we can state:
- \(\vec{R} \cdot \vec{P} = 0\) (indicating \(\vec{R}\) is perpendicular to \(\vec{P}\))
- \(\vec{R} \cdot \vec{Q} = 0\) (indicating \(\vec{R}\) is perpendicular to \(\vec{Q}\))

3. **Analyzing the Statement about \(\vec{P} + \vec{Q}\)**:
- The vector \(\vec{P} + \vec{Q}\) lies in the same plane as \(\vec{P}\) and \(\vec{Q}\).
- Since \(\vec{R}\) is perpendicular to both \(\vec{P}\) and \(\vec{Q}\), it is also perpendicular to any vector that lies in the plane formed by \(\vec{P}\) and \(\vec{Q}\).

4. **Evaluating the Statements**:
- We need to identify which statement is not true based on the properties discussed.
- The statement that \(\vec{R}\) is perpendicular to \(\vec{P} + \vec{Q}\) is true because \(\vec{R}\) is perpendicular to the plane containing \(\vec{P}\) and \(\vec{Q}\).
- However, the statement that \(\vec{R}\) is perpendicular to itself (i.e., \(\vec{R} \cdot \vec{R} = 0\)) is false, since a vector cannot be perpendicular to itself.

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

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